# Neighborhoods and Manifolds of Immersed Curves

Authors: Andrea C. G. Mennucci
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

We present some fine properties of immersionsℐ:M⟶N between manifolds, with particular attention to the case of immersed curves c:S1⟶ℝn. We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly “up to reparameterization.” We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.


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## Body

## 1. Introduction

In general, letM and N be smooth finite dimensional connected Hausdorff paracompact manifolds without boundary, with dimM≤dimN.This paper studies properties of immersionsℐ:M⟶N that are C1 maps such that Tℐx is full rank at each x.A particular but very interesting case are closed immersed curvesc:S1⟶ℝn that are C1 maps with c′θ≠0 at all θ∈S1, where S1=x∈ℝ2:x=1 be the circle in the plane. They will be called planar when n=2.This paper is mostly devoted to this case (a forthcoming paper [15] will generalize many results in this paper to the general case of immersions ℐ:M⟶N).Immersed planar curvesc:S1⟶ℝ2 have been used in computer vision for decades; indeed, the boundary of an object in an image can be modeled as a closed embedded curve, by the Jordan Theorem. Possibly, the first occurrence was active contours, introduced by [9] and used for the segmentation problem: the idea is to minimize energy, defined on contours or curves, that contains an image-based edge attraction term and a smoothness term, which becomes large when the curve is irregular. An evolution is derived to minimize the energy based on principles from the calculus of variations. There have been many variations to original model of [9]; for example [6] and a survey in [3].An unjustified feature of the model of [9] was that the evolution is dependent on the way the contour is parameterized. Thereafter, the authors of [4, 11] considered minimizing a geometric energy, which is a generalization of Euclidean arclength, defined on curves for the edge-detection problem. The authors derived the gradient descent flow in order to minimize the geometric energy.This lead to a principle: all operations related to curves should beindependent of the choice of parameterizations.Operations on the space of curves are best described and studied if the whole space of curves is endowed with a differential structure, so that it becomes a smooth manifold.The above two remarks lead to the following question. IfI is the space of curves that we are interested in and D is the action of reparameterization, then the quotient(1)B=ID,is the space of curves up to parameterization (also called geometric curves in the following): when (and how) can we say that this quotient B is a smooth manifold?This was discussed in [16], using a result from [5].A purpose of this paper is to revisit the key result in [5]: indeed the proof in that paper is missing two key steps.


### 1.1. Plan of the Paper

In Section2, we will define the needed topologies on the space of functions; we will present well known definitions and notations for curves, such as derivation and integration in arc parameter, length, normal vectors, and curvature; we will classify immersed and freely immersed curves and present results and examples.In Section3, we will present advanced results for immersed curves; we will discuss representation of nearby curves in tubular coordinates; we will show how the open neighborhood of a curve c in the space of curves can be defined using tubular coordinates, so that if c is immersed (respectively, freely immersed) then all curves in the neighborhood are immersed (respectively, freely immersed); we will show with examples what goes wrong when hypotheses are not met.In Section4, we will present the proof of this theorem: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold. We will then explain, in a step by step analysis, why the original proof in [5] was incorrect.A supplemental file containsWolfram Mathematica code to generate some of the figures.


## 1.1. Plan of the Paper

In Section2, we will define the needed topologies on the space of functions; we will present well known definitions and notations for curves, such as derivation and integration in arc parameter, length, normal vectors, and curvature; we will classify immersed and freely immersed curves and present results and examples.In Section3, we will present advanced results for immersed curves; we will discuss representation of nearby curves in tubular coordinates; we will show how the open neighborhood of a curve c in the space of curves can be defined using tubular coordinates, so that if c is immersed (respectively, freely immersed) then all curves in the neighborhood are immersed (respectively, freely immersed); we will show with examples what goes wrong when hypotheses are not met.In Section4, we will present the proof of this theorem: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold. We will then explain, in a step by step analysis, why the original proof in [5] was incorrect.A supplemental file containsWolfram Mathematica code to generate some of the figures.


## 2. Definitions

In this section, we will present well-known definitions and results regarding immersions, with particular attention to immersed curves.


### 2.1. Topologies

Definition 1.
We denote byCrM,N the space of all maps f:M⟶N that are of class Cr. Here, r∈0,1,2,…,∞.There are classically two types of topologies for this space.(i)
Theweak topology, as defined in Ch. 1 Sec. 1 in [8], that coincides with the compact-open Cr-topology as defined in 41.9 in [12]; if N=ℝn, then the “weak topology” is the topology of the Fréchet space of local uniform convergence of functions and their derivatives up to order r(ii)
Thestrong topology as defined Ch. 1 Sec. 1 in [8] coincides with the Whitney Cr-topology as defined in 41.10 in [12]IfM is compact, then the two above coincide; if moreover N=ℝn and r<∞, then CrM,ℝn is the usual Banach space.Remark 1.
IfN=ℝn but M is not compact, then “strong topology” does not make CrM,ℝn a topological vector space since it has uncountably many connected components; but the connected component containing f≡0 contains only compactly supported functions, and it has the topology CcrM,ℝn (as defined in 6.9 in Rudin [18]) that is the strict inductive limit (for the definition of strict inductive limit and its properties, we refer to 17G at page 148 in [10]) of the immersions(2)VK⟶CcrM,ℝn,where for each K⊂M compact, VK is the space of f:M⟶N that are zero outside of K, with a standard Banach (or Fréchet, for r=∞) structure.Proposition 1.
The sets of immersions, submersions, and embeddings are open inCrM,N with the strong topology, for r≥1.Proofs are in Ch. 1 Sec. 1 in [8].Definition 2.
Forr∈1,…,∞, let DiffrM be the family of diffeomorphisms of M: all the maps ϕ:M⟶M that are Cr and invertible, and the inverse ϕ−1 is Cr. It is a group, the group operation being “composition of functions.”Proposition 2.
DiffrM is open in CrM,M with the strong topology.See Thm. 1.7 in Ch. 1 Sec. 1 in [8].We will omit the superscript “r” from DiffrM in the following, for ease of notation.


### 2.2. Immersions

#### 2.2.1. Free Immersion

Definition 3.
An immersionℐ:M⟶N is called “free” if ℐ≡ℐ∘ϕ for ϕ∈DiffM implies that ϕ is the identity.Proposition 3 (see [5] Lemma 1.3).
Ifℐ is immersed and ℐϕt=ℐt for all t and ϕt=t for a t, then ϕ=Id.Proof.
Indeed, it is easily seen that(3)t∈M:ϕt=t,is closed; and it is also open, since an immersion is also a local diffeomorphism with its image.As a corollary, ifℐ∘ϕ≡ℐ and ℐ∘ψ≡ℐ and ϕt=ψt for a t, then ϕ≡ψ. Another corollary states the following.Corollary 1 (see [5] Lemma 1.4).
Ifℐ is an immersion and there is a x∈ℝn s.t. ℐt=x for one and only one t, then ℐ is a free immersion.This implies that, whendimM<dimN, the free immersions are a dense subset of all immersions (for all the topologies considered in this paper).


#### 2.2.2. Reparameterizations and Isotropy Group

We first consider the general case of immersionsℐ:M⟶N.Definition 4.
The isotropy group (a.k.a. “stabilizer subgroup” or “little group”)Gℐ is the set of all ϕ∈DiffM such that ℐ≡ℐ∘ϕ; it is a subgroup of DiffM.Obviously,ℐ is freely immersed if and only if Gℐ contains only the identity.We will prove thatGℐ is discrete and finite when M is compact.Remark 2.
If we reparameterizeℐ˜=ℐ∘φ˜, then G1 changes by conjugation:(4)ϕ∈Gℐ⟷φ˜−1∘ϕ∘φ˜∈Gℐ˜.Remark 3.
IfM is orientable, then DiffM has a subgroup Diff+M of orientation preserving diffeomorphisms; for the case of curves, then we obtain that DiffS1 has two connected components(5)DiffS1=Diff+S1∪Diff−S1,where(i)
Diff+S1is the family of diffeomorphisms withϕ′>0, and is a normal subgroup(ii)
Diff−S1is the family of diffeomorphisms withϕ′<0Consider a curvec and let Gc be its isotropy group: we will prove in Lemma 3 that if ϕ∈Gc, then ϕ∈Diff+S1.We will mostly useDiff+S1 in the following.Note thatDiff+S1 is a perfect group [19] (see [13] for a self-contained presentation); it is also a simple group: see Discussion in Sec. 2 in [2] for further references. (The author thanks Prof. Kathryn Mann for her help on these subjects.)


### 2.3. Curves

Remember thatS1=x∈ℝ2:x=1 is the circle in the plane. We will often associate ℝ2=ℂ, for convenience. In this case, we will associate S1=eit,t∈ℝ⊂ℂ.Definition 5.
A closed curve is a mapc:S1⟶ℝn. We will always assume that the curve is of class C1 (at least). The image of the curve, or trace of the curve, is cS1.When convenient, we will (equivalently) viewS1 as ℝ/2π (that is, ℝ modulus 2π translations), and consequently a closed curve will be a map c:ℝ⟶ℝn that is 2π-periodic.In particular, this will be the correct interpretation when we will write the operationθ1+θ2 for θ1,θ2∈S1.Remark 4.
The “distance” of points inS1 will be the intrinsic distance; this distance will be represented by the notation:(6)dS1θ1,θ2,for θ1,θ2∈S1, and it is the length of the shortest arc in S1 connecting the two points θ1,θ2. Note that if we identify S1 to ℝ/2π and pick two points θ1,θ2∈ℝ/2π and represent them as real numbers, it may happen that(7)θ1−θ2>dS1θ1,θ2.Definition 6.
(basepoint). We will select a distinguished pointθ0 in the circle S1: for S1⊂ℝ2, it will be θ0=1,0; for S1⊂ℂ, it will be θ0=1; for S1=ℝ/2π, it will be θ0=0.
Given a curve as above, we will callcθ0 the basepoint for the curve.Example 1 (of a nonfreely immersed curve).
The doubly traversed circle defined as(i)
c2z=z2 for z∈S1 when we consider S1⊂ℂ, or equivalently(ii)
c2θ=cos2θ,sin2θ for θ∈ℝ/2π that we identify with S1
Settingϕt=t+π, we have that c2=c2∘ϕ, so c2 is not freely immersed.Example 2 (taken from [5]).
Note that there are free immersions without a point with only one preimage: consider a “figure eight” which consists of two touching circles. Now, we may map the circle to the figure eight by going first three times around the upper circle, then twice around the lower one. This immersionc:S1⟶ℝ2 is free.We provide a simple Example3 that shows how such curve can be made smooth.


#### 2.3.1. Length, Tangent, and Curvatures

In the following, letc:S1⟶ℝn be an immersed curve.Definition 7.
If the curvec is immersed, we can define the derivation with respect to the arc parameter(8)∂∂s=1c′∂∂θ.We will write∂/∂c instead of ∂/∂s when we are dealing with multiple curves, and we will want to specify which curve is used.Definition 8.
We define the tangent vector(9)Tθ=c′θc′θ=∂∂scθ.Definition 9.
The length of the curvec is(10)lenc=def∫S1c′θdθ.Definition 10.
We define the integration by arc-parameter of a functiong:S1⟶ℝn along the curve c by(11)∫cgsds=def∫S1gθc′θdθ.There are two different definitions ofcurvature of an immersed curve: mean curvatureH and signed curvatureκ, which is defined when c is valued in ℝ2.H and k are extrinsic curvatures, they are properties of the embedding of c into ℝn.Definition 11 (H).
Ifc is C2 regular and immersed, we can define the (mean) curvature H of c as(12)H=∂∂s∂∂sc=∂∂sT.It is easy to prove thatH⊥T.Definition 12 (N).
When the curvec is planar, we can define a normal vector N to the curve, by requiring that N=1, N⊥T and N is rotated π/2 degree anticlockwise with respect to T.Definition 13.
(κ) If c is in ℝ2 and C2, then we can define a signed scalar curvature κ=H,N, so that(13)∂∂sT=κN=H,∂∂sN=−κT.There is a choice of sign in the above two definitions; this choice agrees with the choice in [21].When we will be dealing with multiple curves, we will specify the curve as a subscript,e.g., Tc,κc,Nc will be the tangent, curvature, and normal to the curve c.Remark 5.
Note thatT,κ,N,H are geometrical quantities. If ψ∈Diff+S1 and c˜=c∘ψ, then Tc˜=Tc∘ψ, κc˜=κc∘ψ, Nc˜=Nc∘ψ and Hc˜=Hc∘ψ.


#### 2.3.2. Arc Parameter

Letc be an immersed planar curve. We recall this important transformation.Lemma 1 (constant speed reparameterization).
A curvec∈C1 can be reparameterized to c˜=c∘φ using a φ∈Diff+S1 so that c˜′≡ℓ where ℓ=lenc/2π is constant.Proof.
For simplicity, we assume thatS1=0,2π. Let L=lenc, let ψt=2π/L∫0tc′θ|dθ. Then ψ:0,2π⟶0,2π is a diffeomorphism, let φ=ψ−1.Reparameterization to constant speed is a smooth operation in the space of curves, see Theorem 7 in [20].Whenc′≡1, we will say that the curve is by arc parameter. A curve can be reparameterized to arc parameter without changing its domain (as done above) iff lenc=2π (if this is not the case, we will rescale the curve to make it so.)


#### 2.3.3. Angle Function and Rotation Index

Proposition 4 (angle function and rotation index).
Ifc∈C1 is planar and is immersed, then T=c′/c′ is continuous and T=1, so there exists a continuous function α:ℝ⟶ℝ satisfying(14)Ts=cosαs,sinαs,and αs is unique, up to adding the constant k2π with k∈ℤ. α is called the angle function.
Moreoverαs+2π−αs=2πI, where I is an integer, known as rotation index of c. This number is unaltered if c is deformed by a smooth homotopy (Figure 1).Figure 1
Examples of curves of different rotation index.See 2.1.4 in [1] or Theorem 53.1 in [17], and following.Remark 6.
We can use the angle function to compute the scalar curvatureκ, that was defined in Definition 13 by ∂sT=κN, indeed deriving (14) and combining this with(15)Ns=−sinαs,cosαs,we obtain(16)κ=α′c′=∂sα.


### 2.4. Shapes

Shapes are usually considered to begeometric objects. Representing a curve using c:S1⟶ℝn forces a choice of parameterization that is not really part of the concept of “shape.”Suppose thatI is a space of immersed curves c:S1⟶ℝn.Definition 14 (geometric curves).
The quotient spaceB=I/DiffS1 is the space of curves up to reparameterization, also called geometric curves in the following. Two parametric curves c1,c2∈M such that c1=c2∘ϕ for a ϕ∈DiffS1 are the same geometric curve inside B.B is mathematically defined as the set B=c of all equivalence classesc of curves that are equal but for reparameterization,(17)c=defc∘ϕ,forϕ∈DiffS1.We may also consider the quotient w.r.tDiff+S1. The quotient space I/Diff+S1 is the space of geometric-oriented curves.Unfortunately, the quotient of immersed curves by reparameterizations is not a manifold; but the quotient offreely immersed curve is.Theorem 1.
Suppose thatI is the space of the freely immersed curves; and that I and DiffS1 have the topology of the Fréchet space of C∞ functions, then the quotient B=I/Diff+S1 is a smooth manifold modeled on C∞.One aim of this paper will be to give a complete proof of this result, first presented in [5]. (We remark that the theorem in [5] was presented for the case of immersions ℐ:M⟶N). The proof is in Section 4.1. Indeed, as we will discuss in Section 4.2, the proof in [5] misses some key arguments.


## 2.1. Topologies

Definition 1.
We denote byCrM,N the space of all maps f:M⟶N that are of class Cr. Here, r∈0,1,2,…,∞.There are classically two types of topologies for this space.(i)
Theweak topology, as defined in Ch. 1 Sec. 1 in [8], that coincides with the compact-open Cr-topology as defined in 41.9 in [12]; if N=ℝn, then the “weak topology” is the topology of the Fréchet space of local uniform convergence of functions and their derivatives up to order r(ii)
Thestrong topology as defined Ch. 1 Sec. 1 in [8] coincides with the Whitney Cr-topology as defined in 41.10 in [12]IfM is compact, then the two above coincide; if moreover N=ℝn and r<∞, then CrM,ℝn is the usual Banach space.Remark 1.
IfN=ℝn but M is not compact, then “strong topology” does not make CrM,ℝn a topological vector space since it has uncountably many connected components; but the connected component containing f≡0 contains only compactly supported functions, and it has the topology CcrM,ℝn (as defined in 6.9 in Rudin [18]) that is the strict inductive limit (for the definition of strict inductive limit and its properties, we refer to 17G at page 148 in [10]) of the immersions(2)VK⟶CcrM,ℝn,where for each K⊂M compact, VK is the space of f:M⟶N that are zero outside of K, with a standard Banach (or Fréchet, for r=∞) structure.Proposition 1.
The sets of immersions, submersions, and embeddings are open inCrM,N with the strong topology, for r≥1.Proofs are in Ch. 1 Sec. 1 in [8].Definition 2.
Forr∈1,…,∞, let DiffrM be the family of diffeomorphisms of M: all the maps ϕ:M⟶M that are Cr and invertible, and the inverse ϕ−1 is Cr. It is a group, the group operation being “composition of functions.”Proposition 2.
DiffrM is open in CrM,M with the strong topology.See Thm. 1.7 in Ch. 1 Sec. 1 in [8].We will omit the superscript “r” from DiffrM in the following, for ease of notation.


## 2.2. Immersions

### 2.2.1. Free Immersion

Definition 3.
An immersionℐ:M⟶N is called “free” if ℐ≡ℐ∘ϕ for ϕ∈DiffM implies that ϕ is the identity.Proposition 3 (see [5] Lemma 1.3).
Ifℐ is immersed and ℐϕt=ℐt for all t and ϕt=t for a t, then ϕ=Id.Proof.
Indeed, it is easily seen that(3)t∈M:ϕt=t,is closed; and it is also open, since an immersion is also a local diffeomorphism with its image.As a corollary, ifℐ∘ϕ≡ℐ and ℐ∘ψ≡ℐ and ϕt=ψt for a t, then ϕ≡ψ. Another corollary states the following.Corollary 1 (see [5] Lemma 1.4).
Ifℐ is an immersion and there is a x∈ℝn s.t. ℐt=x for one and only one t, then ℐ is a free immersion.This implies that, whendimM<dimN, the free immersions are a dense subset of all immersions (for all the topologies considered in this paper).


### 2.2.2. Reparameterizations and Isotropy Group

We first consider the general case of immersionsℐ:M⟶N.Definition 4.
The isotropy group (a.k.a. “stabilizer subgroup” or “little group”)Gℐ is the set of all ϕ∈DiffM such that ℐ≡ℐ∘ϕ; it is a subgroup of DiffM.Obviously,ℐ is freely immersed if and only if Gℐ contains only the identity.We will prove thatGℐ is discrete and finite when M is compact.Remark 2.
If we reparameterizeℐ˜=ℐ∘φ˜, then G1 changes by conjugation:(4)ϕ∈Gℐ⟷φ˜−1∘ϕ∘φ˜∈Gℐ˜.Remark 3.
IfM is orientable, then DiffM has a subgroup Diff+M of orientation preserving diffeomorphisms; for the case of curves, then we obtain that DiffS1 has two connected components(5)DiffS1=Diff+S1∪Diff−S1,where(i)
Diff+S1is the family of diffeomorphisms withϕ′>0, and is a normal subgroup(ii)
Diff−S1is the family of diffeomorphisms withϕ′<0Consider a curvec and let Gc be its isotropy group: we will prove in Lemma 3 that if ϕ∈Gc, then ϕ∈Diff+S1.We will mostly useDiff+S1 in the following.Note thatDiff+S1 is a perfect group [19] (see [13] for a self-contained presentation); it is also a simple group: see Discussion in Sec. 2 in [2] for further references. (The author thanks Prof. Kathryn Mann for her help on these subjects.)


## 2.2.1. Free Immersion

Definition 3.
An immersionℐ:M⟶N is called “free” if ℐ≡ℐ∘ϕ for ϕ∈DiffM implies that ϕ is the identity.Proposition 3 (see [5] Lemma 1.3).
Ifℐ is immersed and ℐϕt=ℐt for all t and ϕt=t for a t, then ϕ=Id.Proof.
Indeed, it is easily seen that(3)t∈M:ϕt=t,is closed; and it is also open, since an immersion is also a local diffeomorphism with its image.As a corollary, ifℐ∘ϕ≡ℐ and ℐ∘ψ≡ℐ and ϕt=ψt for a t, then ϕ≡ψ. Another corollary states the following.Corollary 1 (see [5] Lemma 1.4).
Ifℐ is an immersion and there is a x∈ℝn s.t. ℐt=x for one and only one t, then ℐ is a free immersion.This implies that, whendimM<dimN, the free immersions are a dense subset of all immersions (for all the topologies considered in this paper).


## 2.2.2. Reparameterizations and Isotropy Group

We first consider the general case of immersionsℐ:M⟶N.Definition 4.
The isotropy group (a.k.a. “stabilizer subgroup” or “little group”)Gℐ is the set of all ϕ∈DiffM such that ℐ≡ℐ∘ϕ; it is a subgroup of DiffM.Obviously,ℐ is freely immersed if and only if Gℐ contains only the identity.We will prove thatGℐ is discrete and finite when M is compact.Remark 2.
If we reparameterizeℐ˜=ℐ∘φ˜, then G1 changes by conjugation:(4)ϕ∈Gℐ⟷φ˜−1∘ϕ∘φ˜∈Gℐ˜.Remark 3.
IfM is orientable, then DiffM has a subgroup Diff+M of orientation preserving diffeomorphisms; for the case of curves, then we obtain that DiffS1 has two connected components(5)DiffS1=Diff+S1∪Diff−S1,where(i)
Diff+S1is the family of diffeomorphisms withϕ′>0, and is a normal subgroup(ii)
Diff−S1is the family of diffeomorphisms withϕ′<0Consider a curvec and let Gc be its isotropy group: we will prove in Lemma 3 that if ϕ∈Gc, then ϕ∈Diff+S1.We will mostly useDiff+S1 in the following.Note thatDiff+S1 is a perfect group [19] (see [13] for a self-contained presentation); it is also a simple group: see Discussion in Sec. 2 in [2] for further references. (The author thanks Prof. Kathryn Mann for her help on these subjects.)


## 2.3. Curves

Remember thatS1=x∈ℝ2:x=1 is the circle in the plane. We will often associate ℝ2=ℂ, for convenience. In this case, we will associate S1=eit,t∈ℝ⊂ℂ.Definition 5.
A closed curve is a mapc:S1⟶ℝn. We will always assume that the curve is of class C1 (at least). The image of the curve, or trace of the curve, is cS1.When convenient, we will (equivalently) viewS1 as ℝ/2π (that is, ℝ modulus 2π translations), and consequently a closed curve will be a map c:ℝ⟶ℝn that is 2π-periodic.In particular, this will be the correct interpretation when we will write the operationθ1+θ2 for θ1,θ2∈S1.Remark 4.
The “distance” of points inS1 will be the intrinsic distance; this distance will be represented by the notation:(6)dS1θ1,θ2,for θ1,θ2∈S1, and it is the length of the shortest arc in S1 connecting the two points θ1,θ2. Note that if we identify S1 to ℝ/2π and pick two points θ1,θ2∈ℝ/2π and represent them as real numbers, it may happen that(7)θ1−θ2>dS1θ1,θ2.Definition 6.
(basepoint). We will select a distinguished pointθ0 in the circle S1: for S1⊂ℝ2, it will be θ0=1,0; for S1⊂ℂ, it will be θ0=1; for S1=ℝ/2π, it will be θ0=0.
Given a curve as above, we will callcθ0 the basepoint for the curve.Example 1 (of a nonfreely immersed curve).
The doubly traversed circle defined as(i)
c2z=z2 for z∈S1 when we consider S1⊂ℂ, or equivalently(ii)
c2θ=cos2θ,sin2θ for θ∈ℝ/2π that we identify with S1
Settingϕt=t+π, we have that c2=c2∘ϕ, so c2 is not freely immersed.Example 2 (taken from [5]).
Note that there are free immersions without a point with only one preimage: consider a “figure eight” which consists of two touching circles. Now, we may map the circle to the figure eight by going first three times around the upper circle, then twice around the lower one. This immersionc:S1⟶ℝ2 is free.We provide a simple Example3 that shows how such curve can be made smooth.


### 2.3.1. Length, Tangent, and Curvatures

In the following, letc:S1⟶ℝn be an immersed curve.Definition 7.
If the curvec is immersed, we can define the derivation with respect to the arc parameter(8)∂∂s=1c′∂∂θ.We will write∂/∂c instead of ∂/∂s when we are dealing with multiple curves, and we will want to specify which curve is used.Definition 8.
We define the tangent vector(9)Tθ=c′θc′θ=∂∂scθ.Definition 9.
The length of the curvec is(10)lenc=def∫S1c′θdθ.Definition 10.
We define the integration by arc-parameter of a functiong:S1⟶ℝn along the curve c by(11)∫cgsds=def∫S1gθc′θdθ.There are two different definitions ofcurvature of an immersed curve: mean curvatureH and signed curvatureκ, which is defined when c is valued in ℝ2.H and k are extrinsic curvatures, they are properties of the embedding of c into ℝn.Definition 11 (H).
Ifc is C2 regular and immersed, we can define the (mean) curvature H of c as(12)H=∂∂s∂∂sc=∂∂sT.It is easy to prove thatH⊥T.Definition 12 (N).
When the curvec is planar, we can define a normal vector N to the curve, by requiring that N=1, N⊥T and N is rotated π/2 degree anticlockwise with respect to T.Definition 13.
(κ) If c is in ℝ2 and C2, then we can define a signed scalar curvature κ=H,N, so that(13)∂∂sT=κN=H,∂∂sN=−κT.There is a choice of sign in the above two definitions; this choice agrees with the choice in [21].When we will be dealing with multiple curves, we will specify the curve as a subscript,e.g., Tc,κc,Nc will be the tangent, curvature, and normal to the curve c.Remark 5.
Note thatT,κ,N,H are geometrical quantities. If ψ∈Diff+S1 and c˜=c∘ψ, then Tc˜=Tc∘ψ, κc˜=κc∘ψ, Nc˜=Nc∘ψ and Hc˜=Hc∘ψ.


### 2.3.2. Arc Parameter

Letc be an immersed planar curve. We recall this important transformation.Lemma 1 (constant speed reparameterization).
A curvec∈C1 can be reparameterized to c˜=c∘φ using a φ∈Diff+S1 so that c˜′≡ℓ where ℓ=lenc/2π is constant.Proof.
For simplicity, we assume thatS1=0,2π. Let L=lenc, let ψt=2π/L∫0tc′θ|dθ. Then ψ:0,2π⟶0,2π is a diffeomorphism, let φ=ψ−1.Reparameterization to constant speed is a smooth operation in the space of curves, see Theorem 7 in [20].Whenc′≡1, we will say that the curve is by arc parameter. A curve can be reparameterized to arc parameter without changing its domain (as done above) iff lenc=2π (if this is not the case, we will rescale the curve to make it so.)


### 2.3.3. Angle Function and Rotation Index

Proposition 4 (angle function and rotation index).
Ifc∈C1 is planar and is immersed, then T=c′/c′ is continuous and T=1, so there exists a continuous function α:ℝ⟶ℝ satisfying(14)Ts=cosαs,sinαs,and αs is unique, up to adding the constant k2π with k∈ℤ. α is called the angle function.
Moreoverαs+2π−αs=2πI, where I is an integer, known as rotation index of c. This number is unaltered if c is deformed by a smooth homotopy (Figure 1).Figure 1
Examples of curves of different rotation index.See 2.1.4 in [1] or Theorem 53.1 in [17], and following.Remark 6.
We can use the angle function to compute the scalar curvatureκ, that was defined in Definition 13 by ∂sT=κN, indeed deriving (14) and combining this with(15)Ns=−sinαs,cosαs,we obtain(16)κ=α′c′=∂sα.


## 2.3.1. Length, Tangent, and Curvatures

In the following, letc:S1⟶ℝn be an immersed curve.Definition 7.
If the curvec is immersed, we can define the derivation with respect to the arc parameter(8)∂∂s=1c′∂∂θ.We will write∂/∂c instead of ∂/∂s when we are dealing with multiple curves, and we will want to specify which curve is used.Definition 8.
We define the tangent vector(9)Tθ=c′θc′θ=∂∂scθ.Definition 9.
The length of the curvec is(10)lenc=def∫S1c′θdθ.Definition 10.
We define the integration by arc-parameter of a functiong:S1⟶ℝn along the curve c by(11)∫cgsds=def∫S1gθc′θdθ.There are two different definitions ofcurvature of an immersed curve: mean curvatureH and signed curvatureκ, which is defined when c is valued in ℝ2.H and k are extrinsic curvatures, they are properties of the embedding of c into ℝn.Definition 11 (H).
Ifc is C2 regular and immersed, we can define the (mean) curvature H of c as(12)H=∂∂s∂∂sc=∂∂sT.It is easy to prove thatH⊥T.Definition 12 (N).
When the curvec is planar, we can define a normal vector N to the curve, by requiring that N=1, N⊥T and N is rotated π/2 degree anticlockwise with respect to T.Definition 13.
(κ) If c is in ℝ2 and C2, then we can define a signed scalar curvature κ=H,N, so that(13)∂∂sT=κN=H,∂∂sN=−κT.There is a choice of sign in the above two definitions; this choice agrees with the choice in [21].When we will be dealing with multiple curves, we will specify the curve as a subscript,e.g., Tc,κc,Nc will be the tangent, curvature, and normal to the curve c.Remark 5.
Note thatT,κ,N,H are geometrical quantities. If ψ∈Diff+S1 and c˜=c∘ψ, then Tc˜=Tc∘ψ, κc˜=κc∘ψ, Nc˜=Nc∘ψ and Hc˜=Hc∘ψ.


## 2.3.2. Arc Parameter

Letc be an immersed planar curve. We recall this important transformation.Lemma 1 (constant speed reparameterization).
A curvec∈C1 can be reparameterized to c˜=c∘φ using a φ∈Diff+S1 so that c˜′≡ℓ where ℓ=lenc/2π is constant.Proof.
For simplicity, we assume thatS1=0,2π. Let L=lenc, let ψt=2π/L∫0tc′θ|dθ. Then ψ:0,2π⟶0,2π is a diffeomorphism, let φ=ψ−1.Reparameterization to constant speed is a smooth operation in the space of curves, see Theorem 7 in [20].Whenc′≡1, we will say that the curve is by arc parameter. A curve can be reparameterized to arc parameter without changing its domain (as done above) iff lenc=2π (if this is not the case, we will rescale the curve to make it so.)


## 2.3.3. Angle Function and Rotation Index

Proposition 4 (angle function and rotation index).
Ifc∈C1 is planar and is immersed, then T=c′/c′ is continuous and T=1, so there exists a continuous function α:ℝ⟶ℝ satisfying(14)Ts=cosαs,sinαs,and αs is unique, up to adding the constant k2π with k∈ℤ. α is called the angle function.
Moreoverαs+2π−αs=2πI, where I is an integer, known as rotation index of c. This number is unaltered if c is deformed by a smooth homotopy (Figure 1).Figure 1
Examples of curves of different rotation index.See 2.1.4 in [1] or Theorem 53.1 in [17], and following.Remark 6.
We can use the angle function to compute the scalar curvatureκ, that was defined in Definition 13 by ∂sT=κN, indeed deriving (14) and combining this with(15)Ns=−sinαs,cosαs,we obtain(16)κ=α′c′=∂sα.


## 2.4. Shapes

Shapes are usually considered to begeometric objects. Representing a curve using c:S1⟶ℝn forces a choice of parameterization that is not really part of the concept of “shape.”Suppose thatI is a space of immersed curves c:S1⟶ℝn.Definition 14 (geometric curves).
The quotient spaceB=I/DiffS1 is the space of curves up to reparameterization, also called geometric curves in the following. Two parametric curves c1,c2∈M such that c1=c2∘ϕ for a ϕ∈DiffS1 are the same geometric curve inside B.B is mathematically defined as the set B=c of all equivalence classesc of curves that are equal but for reparameterization,(17)c=defc∘ϕ,forϕ∈DiffS1.We may also consider the quotient w.r.tDiff+S1. The quotient space I/Diff+S1 is the space of geometric-oriented curves.Unfortunately, the quotient of immersed curves by reparameterizations is not a manifold; but the quotient offreely immersed curve is.Theorem 1.
Suppose thatI is the space of the freely immersed curves; and that I and DiffS1 have the topology of the Fréchet space of C∞ functions, then the quotient B=I/Diff+S1 is a smooth manifold modeled on C∞.One aim of this paper will be to give a complete proof of this result, first presented in [5]. (We remark that the theorem in [5] was presented for the case of immersions ℐ:M⟶N). The proof is in Section 4.1. Indeed, as we will discuss in Section 4.2, the proof in [5] misses some key arguments.


## 3. Advanced Properties of Immersed Curves

In this section, we will present results regarding immersed curves that are either new or presented in more precise form than usually found in the literature.Most of the results are presented, for sake of simplicity, for planar curvesc:S1⟶ℝ2, but can be extended to the case of curves c:S1⟶N taking values in a manifold N, up to some nuisance in notations.The general case of immersionsℐ:M⟶N requires instead some arguments that will be discussed in a future paper [15].


### 3.1. Examples

Definition 15.
We start with some classical examples ofC∞ functions of compact support. Let(18)η0:ℝ⟶ℝ,η0u=defe4/π−1/u−1/π−u,u∈0,π,0,elsewhere,(19)η1:ℝ⟶ℝ,η1u=def42u−πe−1/u−1/π−u,u∈0,π,0,elsewhere,(see Figure 2 on the following page).Figure 2
Graph ofη0 as defined in (18); graph of η1 as defined in (19).We will use these to build some following examples.Example 3.
We present here a simple smooth formula for Example2(20)c:0,5π⟶ℝ2,cu=defsin2u,c2u,c2uη0u,u≤π,η0u−π,u≤2π,−η0u−2π,u≤3π,−η0u−3π,u≤4π,−η0u−4π,u≤5π.this is a C∞ function depicted at Figure 3 on the next page.Figure 3
x and y coordinates of curve defined in Example 2; trace of curve defined in Example 2.


#### 3.1.1. Trace and Parameterization

If a curve is embedded then the curve is identified by its image, in these senses.(i)
Ifc,c˜:S1⟶ℝn are embedded and have the same image, then there is an unique reparameterization ϕ such that c˜=c∘ϕ.(ii)
Suppose thatc0:S1⟶ℝn is embedded and A=cS1 is the trace; suppose that c0 is parameterized by constant speed parameter; let us fix a candidate basepoint v∈A in the trace.We can state thatA,v characterize the embedded curve up to a choice of direction: precisely, there are exactly two different c1,c2:S1⟶ℝn, parameterized by constant speed parameter, such that c1θ0=c2θ0=v, and they satisfy(21)c1θ=c0θ+a,c2θ=c0b−θ,for unique choices of a,b∈S1 (dependant on v).In particular, if the rotation index ofc0 is r, then the latter curves have rotation indexes ±r.Since the definition offreely immersed curve says that the curve identifies a unique parameterization, then we may be induced to think that the above two properties extend to freely immersed curves: but this is not the case.Example 4.
The following two curves have the same trace, are freely immersed, are smooth, but have rotation indexes 0 and 1.(1)
This immersed closed curvec:0,2π⟶ℝ2 with components(22)c1u=sin2u,c2u=η12u,u∈0,π2,η12u−π2,u∈π2,π.(2)
This immersed closed curvec:0,2π⟶ℝ2 with components(23)c2u=sin2u,c2u=η12u,u∈0,π2,−η12u−π2,u∈π2,π.
See Figure4 on the following page.Figure 4
Trace of the two curves and parameterization defined in Example4; with color scheme used for the parameter.


### 3.2. Reparameterizations and Isotropy Group

Lemma 2.
Ifϕ∈Diff−S1, then ϕ has two fixed points.Proof.
We representϕ as a map ϕ:0,2π⟶ℝ that is continuous, strictly decreasing and such that(24)ϕ0∈2π,4π,ϕ2π=ϕ0−2π,then(25)ϕ2π∈0,2π,so the graph y=ϕx must intersect both the graph y=x and the graph y=x+2π for two different points x1,x2∈0,2π that are the two fixed points.Lemma 3.
Ifc is immersed and ψ∈Gc, then ψ∈Diff+S1.Proof.
Suppose thatψ∈Diff−S1, let u∈S1 be a fixed point (by Lemma 2). By deriving(26)c′θ=c′ψθψ′θ,setting θ=u(27)c′u=c′uψ′u,and this is impossible since ψ′u<0.


### 3.3. Local Embedding

#### 3.3.1. Length of Curve Arcs

Definition 16.
Supposec:S1⟶ℝn is C1. Let σ,σ˜∈S1. When σ≠σ˜ there are two arcs in S1 connecting σ to σ˜. By(28)lenc|σ,σ˜,we will mean the minimum of the lengths of c when restricted to one of the two arcs connecting σ to σ˜.Ifc is periodically extended to c:ℝ⟶ℝn and σ,σ˜∈ℝ, then there is an unique k∈ℤ such that(29)σ≤σ˜+k2π<σ+2π,and then, letting(30)l1=∫σσ˜+k2πc′θdθ,l2=∫σ˜+k2πσ+2πc′θdθ,we define(31)lenc|σ,σ˜=minl1,l2.In particular, whenc is parameterized at constant speed (i.e.,c′≡ℓ), then we will (covertly) assume that σandσ˜ are chosen (up to adding k2π) so that σ−σ˜=dS1σ,σ˜≤π and then(32)lenc|σ,σ˜=ℓσ−σ˜.Remark 7.
Whenc is not parameterized by constant velocity, the above may lead to some confusion. The interval σ,σ˜ in the notation (28) implicitly refers to the choice of arc in S1 that provides the above minimum. Note that this may not be the shortest arc connecting σ to σ˜ in S1. This may happen if the parameterization of c has regions of fast and slow velocity, as in this example.Example 5.
Letc˜θ=cosθ,sinθ be the standard circle, and(33)ϕθ=4θ,0≤θ≤π3,25x+3π,π3≤θ≤2π,(see plot in Figure 5) then smooth out the corners of ϕ so that it becomes a diffeomorphism of S1; let(34)c=c˜∘ϕ,let θ0=0,θ1=π/3 in S1∼ℝ/2π, then dS1θ0,θ1=π/3, and is given by the arc moving counterclockwise from θ0 to θ1, while(35)lenc|σ0,σ1=2π3,is given by the arc moving clockwise from θ0 to θ1.Figure 5
Plot of functionϕ in equation (33) in Example 5.This never happens for small distances/lengths, though.Theorem 2.
Fix an immersed curvec:S1⟶ℝn; let(36)M=maxc′,m=minc′,σc=def1M/m+1.(i)
For anyθ0,θ1 in S1 such that(37)dS1θ0,θ1<2πσc,the shortest arc connecting them inS1 is also the arc where(38)lenc|θ0,θ1is computed(ii)
For anyθ0,θ1inS1such that(39)lenc|θ0,θ1<σclencthe arc where(40)lenc|θ0,θ1,is computed is also the shortest arc connecting them inS1, whose length is(41)dS1θ0,θ1.(iii)
In any of the above cases,(42)mdS1θ0,θ1≤lenc|θ0,θ1≤MdS1θ0,θ1.


#### 3.3.2. Estimates

We begin with this estimate.Proposition 5.
Letαs be the angle function, for s∈J=0,2π. The fact that the curve is closed imposes lower bounds on maxJα′.(i)
If the rotation indexℐ of the curve is not zero, then α2π−α0=2πI so necessarily maxJα′≥I(ii)
If the rotation index of the curve is zero, then necessarilymaxJα′≥1/2
Indeed, we can prove that(43)maxJα−minJα≥π;otherwise, let(44)β=maxJα+minJα2,taking v=cosβ,sinβ, we would have(45)v⋅c′θ=c′θcosβ−αθ>0,for all θ, hence the curve would not be closed.Definition 17.
Givenc:S1⟶ℝ2, a C2 immersed closed curve, we recall that κ is the scalar curvature of c; we define (46)δc=defπ3maxκ,τc=def12maxκ.Note that since the curve is closed, κ cannot be identically zero.Note thatδc=τc2π/3, but we define two quantities since this simplifies the notation in the following. We have 2τc≤δc but 3τc≥δc.Remark 8.
Note that if we rescale the curvec by a factor λ, then δc,τc and lenc|σ,σ˜ are multiplied by λ as well. If we rotate or translate c, then δc,τc and lenc|σ,σ˜ are unaffected. If we reparameterize, then δc,τc are unchanged, whereas if ψ∈DiffS1 and c˜=c∘ψ, we have(47)lenc˜|ψσ,ψσ˜=lenc|σ,σ˜.
In all (with the exception of relation (100) in Lemma 7) following definitions, propositions, and theorems, the formulas are built to be “geometrical”: this means that, if the curves are reparameterized, rescaled, translated or rotated, then the formulas change in predictable ways (as explained above).
This simplifies the proofs: in the proofs we can assume, with no loss of generality, that the curve is parameterized by arc parameter.Remark 9.
Note thatδc≤lenc/3 for curves of index zero and δc≤lenc/6I for curves of index I≠0.Proof.
We use Proposition5. The formula in the thesis is invariant for reparameterizations and scaling; we rescale the curve so that lenc=2π and reparameterize by arc parameter so that c′≡1. For curves of index zero, the thesis π/3maxJκ≤lenc/3, that is, π≤maxJκlenc; since c′≡1 by (16), this last becomes 1/2≤maxJα′ that was proved above. For curves of index I, the thesis π/3maxJκ≤lenc/6I, that is, 2πI≤maxJκlenc, then becomes I≤maxJα′ that was proved above.ForI≠0 the above is sharp, as in the case of cθ=cosIθ,sinIθ.


#### 3.3.3. Local Embedding of Curves

It is well known that aC1 immersion ℐ:M⟶N is a local embedding. For curves of class C2, we can provide a simple quantitative statement.Proposition 6 (local embedding).
Letc:S1⟶ℝ2 a C2 immersed curve. Define δc as in Definition 17. For any a,b∈S1, let L=deflenc|a,b; assume that L≤2δc, then cb−ca≥L/2; so, c|a,b is embedded.Proof.
For simplicity, we assume thatc is periodically extended to c:ℝ⟶ℝ2; then, we identify the interval in ℝ that is associated to the arc of the curve where the length lenc|a,b is computed; for simplicity, we call this interval a,b again. (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
Using Lemma1 and Remark 8, assume that c′≡ℓ=lenc/2π; then, ∂s=1/ℓ∂θ, so T=c′/ℓ and(48)lenc|a,b=ℓb−a.
As noted in (16)(49)κ=∂sT=T′ℓ=α′ℓ,so δc=ℓπ/3maxα′. Let m=a+b/2 be the middle point. Let αt be the angle function (14). Up to rotation, suppose c′m=ℓ,0,Tm=1,0; so, we can assume αm=0. Let L=lenc|a,b=ℓb−a≤2δc. For any θ∈a,b, we have ℓθ−m≤δc; hence, θ−m≤π/3maxα′; hence, for all a≤θ≤b, we have(50)αθ≤θ−mmaxα′≤π3,and hence, cosαθ≥1/2; hence, for a≤θ1≤θ2≤b, for the abscissa, we can write(51)c1θ2−c1θ1=ℓ∫θ1θ2cosαθdθ≥ℓθ2−θ112.


### 3.4. Isotropy Group Is Discrete

Given an immersionℐ:M⟶N, it is possible to prove that the isotropy group is discrete (when M is paracompact) and even finite (when M is compact; this latter result appears in [5]). When considering curves, we can obtain the same results (and even more) in a more direct and geometric way.Lemma 4.
Letc:S1⟶ℝn be immersed.(i)
Gcis finite(ii)
Ifc∘ψ≡c∘ϕandψa=ϕafor ana∈S1, thenψ≡ϕ(iii)
Ifc is parameterized by constant speed (see Lemma 1), then there is a k∈ℕ,k≥1 s.t. Gc is the set of all ϕt=t+2πj/k for j=0,…,k−1 (the proof of this is a special case of the 2nd step of the proof of Lemma 9)Proof.
(i)
We prove the third point. Indeed derivingc=c∘ϕ and noting that c′≡ℓ, we obtain ϕ′≡1 so ϕt=t+β; hence, ϕjt=t+jβ∈Gc, for all j; if β/π is irrational, then jβ would be dense in S1=ℝ/2π, and this is denied by Proposition 6. Moreover, if ϕt=t+2π/k and c=c∘ϕ, then c0=c2π/k; but by Proposition 6, 2π/3maxκ≤2πℓ/k, that is, k≤3ℓmaxκ. So, there is an unique k such that any ϕ∈Gc can be written as ϕt=t+2πj/k.(ii)
The above characterization shows that ifϕ′≡1≡ψ′ and ϕa=ψa, then ϕ≡ψ. By Remark 2, this is valid for any curve (even when it is not parameterized by constant speed). This proves the second point.(iii)
The first point follows again from Remark2.


### 3.5. Tubular Neighborhoods

Existence of tubular neighborhood is well known; we provide a quantitative result forC2 planar immersed curves.Proposition 7 (tubular neighborhood).
Defineδc,τc as in Definition 17. Fix a,b∈S1 with lenc|a,b≤2δc. Let(52)Φ:a,b×−τc,τc⟶ℝ2,Φs,t=cs+tNs,then Φ is a diffeomorphism with its image. Moreover, if the arc s1,s2 is contained in the arc a,b identified above, then(53)Φs1,t1−Φs2,t2≥14lenc|s1,s2,whereas (obviously)(54)Φs,t1−Φs,t2=t2−t1.Proof.
Assume that the curve has length2π and is parameterized in arc parameter; with no loss of generality (recalling Remark 8); let α be the angle function (14).
Extendc to a periodic function c:ℝ⟶ℝ2 and identify the interval in ℝ that is associated with the arc of the curve where the length lenc|a,b is computed. For simplicity, we call this interval a,b again (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
The Jacobian ofΦ is(55)∂∂sΦ=T1−κt,∂∂tΦ=N,so its determinant is 1−κt≥1/2 by the hypothesis t≤τc.
We will then prove thatΦ is injective, so it will be an homeomorphism with its image, and since it is a local diffeomorphism, it will be a diffeomorphism.
Chooses1,t1 and s2,t2 with a≤s1<s2≤b and t1≤τc, t2≤τc.
We setm=s2+s1/2. Up to rotation, we assume that Tm=e1=1,0 and αm=0, so that Nm is perpendicular to the x axis. As in Proposition 6, we can prove that cosαs≥1/2 for all s1≤s≤s2.
We write(56)Φs,t=cs+tNs=cm+∫msTsdθ+tNs,and then for the abscissa(57)Φs,t1=cm1+∫mscosαθdθ+tsinαs,note that Φm,t1=cm1. Deriving, we obtain(58)∂∂sΦs,t1=cosαs1−tα′s≥14.
We then obtain that(59)Φs2,t21−Φm,t21≥14s2−m,while(60)Φm,t11−Φs1,t11≥14m−s1,and recalling that Φm,t11=Φm,t21=cm1 and summing, we obtain(61)Φs2,t21−Φs1,t11≥14s2−s1.We will call tubular coordinates aroundc the formula (52).


#### 3.5.1. Counterexample

The hypothesis “c∈C2” in the previous proposition may be broadened to “c∈C1,1”; but the results fails if we only assume that “c∈C1,α” with α∈0,1, as seen in this example (adapted from [1]).Example 6.
Let0<α<1 and cθ=θ,θ1+α; then, for θ>0,(62)cθ+aNθ=θ−aα+1θαα+12θ2α+1,aα+12θ2α+1+θα+1,and this meets the y axes for(63)a=a˜θ=θ1−αα+12θ2α+1α+1.
So, by symmetry,(64)cθ+a˜θNθ=c−θ+a˜−θN−θ,and at the same time,(65)limθ⟶0a˜θ=0.


#### 3.5.2. Nearby Points

Supposec:S1⟶ℝ2 is a C2 immersed curve. Let 0<τ≤τc and Vτ=S1×−τ,τ and(66)Φ:Vτ⟶ℝ2,Φs,t=cs+tNs,and Uτ=ΦVτ.Proposition 8.
Uτ is also the set of points at distance at most τ from the trace cS1.Proof.
LetK=cS1 be the trace of the curve (it is a compact subset of ℝ2). We use the distance function dK:ℝ2⟶ℝ defined as(67)dKx≔infy∈Kx−y,(for an introduction to this object, see [14] and references therein).
Letx∈Uτ, there is a θ,t∈Vτ such that(68)x=cθ+tNθ.
So,(69)x−cθ≤t≤τ,then let θ˜ be a minimum for(70)x−cθ.
So, clearly,(71)dKx=x−cθ˜≤τ.
Vice versa ifdKx≤τ let θ˜ be a minimum as above, then geometrical considerations tell that the segment from x to cθ˜ is orthogonal to the tangent Tθ˜ at cθ˜.As a corollary of Proposition7, for any such neighborhood of the image of c, the “projection to c” is a C1 multivalued map (with finitely many projections in S1).


### 3.6. Not a Covering Map

By looking at the previous Proposition7, we may think that Φ is the universal covering map of U=ΦV (see [17] for the definition). This would be very convenient, and indeed, we could use the lifting lemma to ease some of the following proofs.Supposec˜,c:S1⟶ℝ2 are C2 immersed curves. Consider this statement, that is usually called lifting lemma:If the trace ofc˜ is contained in Uτ, then there is a choice of continuous φ:ℝ⟶ℝ, a:ℝ⟶−τ,τ such that(72)c˜θ=Φφθ,aθ=cφθ+aθNφθ,∀θ.Unfortunately, this is not the case, as seen in this example in Figure6, where the curve c is blue and the curve c˜ is red. The trace of the curve c˜ is all contained in the open set Uτc, but representation (72) cannot hold. We can though prove a version of the lifting lemma useful in the following.Figure 6
Figure for example in Section3.6; the curve c is blue and the curve c˜ is red.


### 3.7. Neighborhoods

#### 3.7.1. Nearby Projection

Lemma 5 (nearby projection).
Fix aCR immersed curve c, with R≥2.(1)
Ifx∈ℝ2 and σ˜∈S1 and(73)d=defx−cσ˜<δc4,then there is ana∈ℝ with a≤d and a σ∈S1 with(74)lenc|σ,σ˜≤4d,such that(75)x=cσ+aNσ.Note also thata is uniquely identified by σ.(2)
They are unique in the family ofσ,a such that a≤τc and(76)lenc|σ,σ˜≤δc,so we can seeσ,a as functions of x, as follows.(3)
Considerx˜∈ℝ2 and σ˜∈S1 for which(77)d=defx˜−cσ˜≤τc2,let ε>0 small such that d+ε<δc/4 and let B=Bx˜,ε=x∈ℝ2:x−x˜<ε for convenience. There is a choice of function a,φ:B⟶ℝ of class CR−1 such that(78)x=Φφx,ax=cφx+axNφx,for all x∈B, and they are unique as specified above.
Note thatτc/2<δc/4<τc.Proof.
Supposec is by arc parameter (with no loss of generality as explained in Remark 8); so we write (recalling (32)) a−b instead of lenc|a,b.(i)
Choosex,σ˜ as in the statement and let Jσ˜=σ˜−δc,σ˜+δc, then consider any minimum point θ^ of(79)minθ∈Jσ˜x−cθ,note that the minimum value has to be less thand; so,(80)cσ˜−cθ^≤cσ˜−x+x−cθ^≤2d,but at the same time (sinceσ˜ and θ^ are at arc distance at most δc), by the previous Proposition 6.(81)cσ˜−cθ^≥12θ^−σ˜.So, combining the two(82)4d≥θ^−σ˜,but4d<δc so θ^ is not at extremes. Then, any θ^ providing the minimum must be internal in the interval Jσ˜: by geometrical reasoning the segment from x to cθ^ is orthogonal to the curve so there is a a such that(83)x=Φθ^,a.(ii)
Recall thatδc/4≤τc; the map Φ is injective for θ∈Jσ˜ and a≤τc, so θ^,a are unique.(iii)
For anyx∈B, we have(84)x−cσ˜<d+ε,and since d+ε<δc/4, then there is an unique σ∈Jσ˜ and a with a≤τc such that(85)x=cσ+aNσ,and we denote them by σ=φx,a=ax. Moreover, we can invert the function(86)Φ:Jσ˜×−τc,τc⟶ℝ2,and write(87)φx,ax=Φ−1x,for x∈B. This proves that φ,a∈C1B.


#### 3.7.2. Global Lifting

Proposition 9 (global lifting).
Supposec:S1⟶ℝ2 is a CR immersed curve and c˜:S1⟶ℝ2 is CR−1, with R≥2. Fix 0<τ<δc/4. Suppose that we have c˜θ−cθ≤τ for all θ. There exists choice of a:S1⟶ℝ and φ:S1⟶S1 such that(88)c˜σ=Φφσ,aσ=cφσ+aσNcφσ,∀σ∈S1,with aσ≤τ and(89)lenc|σ,φσ≤4τ,holding for all σ. And, they are unique in the class of CR−1 functions such that a≤τc and(90)lenc|σ,φσ≤δc.Note also thata is uniquely identified by φ.Proof.
We just substitutex=c˜σ in the previous Lemma. By the second point, we can define functions φθandaθ uniquely as prescribed. By the third point, they are C1.Remark 10.
Suppose we are given two curvesc1,c2 and we know that there exists a choice a,φ such that(91)c2σ=c1φσ+aσNc1φσ,∀σ∈S1,as in (88). If we rotate or translate the two curves, then the above relation will hold, with the same a,φ. If we rescale the two curves by λ>0, then the relation will hold with λa,φ.
If we chooseψ∈Diff+S1 and we reparameterize all curves at the same time by c˜i=ci∘ψ, then(92)c˜2s=c˜1φ˜s+a˜sNc˜1φ˜s,∀s∈S1,holds for(93)a˜=a∘ψ,φ˜=ψ−1∘φ∘ψ.
This follows from direct computation and Remark5.Example 7.
So far so good, butφ may fail to be a diffeomorphism, as in this simple example in Figure 7. where the curve c is blue and the curve c˜ is red.But some simple lemmas can help.Figure 7
Curves in Example7; curve c is blue and curve c˜ is red.Figure 8
Scheme for Lemma 6.Lemma 6.
Let0<α<1. If we have w,v∈ℝn such that(94)w−v≤αv,then the angle β between v and w satisfies β≤arcsinα, and moreover,(95)1−αv≤v⋅w≤1+αv,1−αv≤w≤1+αv.See Figure 8 on the next page.Remark 11.
Letα>0. Suppose c1,c2,c3:S1⟶ℝ2 are C1 maps and(96)c2′θ−c3′θ≤αc1′θ,if ψ∈Diff+S1 and we reparameterize all curves at the same time by c˜i=ci∘ψ, then(97)c˜2′θ−c˜3′θ≤αc˜3′θ.
Similarly, if we rescale, rotate, or translate all curves at the same time.So, (96) is a geometric estimate, indeed we may rewrite it as(98)∂c1c2θ−∂c1c3θ≤α,where ∂c1=1/c1′∂/∂θ.Lemma 7.
Assume all hypotheses in Proposition 9. Assume moreover thatτ≤τc/4 and assume that(99)c˜′σ−c′σ≤12c′σ,∀σ∈S1,then c˜ is immersed and φ is a diffeomorphism. Moreover, when c is parameterized at constant speed, we can state that(100)15≤φ′≤4.Proof.
We rescale and reparameterizec to arc parameter using a reparameterization ψ, and at the same time, we rescale and reparameterize c˜ using the same rescaling and ψ (note that c˜ is not necessarily by arc parameter); with no loss in generality, as explained in Remarks 10 and 11.
Letβ be the angle between c˜′σ and c′σ: by Lemma 6, β=arcsinα, so it is at most π/6. Let θ=φσ; we know that σ−θ≤4τ≤τc; the angle γ between c′σ and c′θ is at most σ−θmaxk so γ≤1/2: so the angle β+γ between c˜′σ and c˜′φσ is at most 1/2+π/6, and this is less than π/2.
Deriving inσ and assuming that c is by arc parameter(101)c˜′σ=Tφ′σ1−κa+a′N,where T,N,κ are evaluated at φσ; then,(102)c˜′⋅T=φ′σ1−κa,now if a≤τc then 1−κa≥1/2; moreover, by the above reasoning c˜′⋅T>0, so φ′σ>0. Moreover, we note that 1/2≤c˜′≤3/2, 1/2≤1−κa≤3/2 and cos1/2+π/6≤6/10 to prove (100). Relation (93) tells then that φ will always be a diffeomorphism, for any curve satisfying the hypotheses.We summarize all the above: we show sufficient hypothesis such thatc˜ may be represented intubular coordinates aroundc.Theorem 3 (representation theorem).
Supposec:S1⟶ℝ2 is a CR immersed curve and c˜:S1⟶ℝ2 is CR−1, with R≥2. Define δc,τc as in Definition 17. Fix 0<τ≤τc/4. Suppose that we have c˜θ−cθ≤τ and(103)c˜′θ−c′θ≤c′θ2,for all θ.
Then,c˜ is immersed, there are φ∈Diff+S1 and a:S1⟶−τ,τ of class CR−1 such that(104)c˜φθ=Φθ,aθ=cθ+aθNθ,∀θ,with aσ≤τ and(105)lenc|σ,φσ≤4τ,holding for all σ.
They are unique in the class ofC1 functions such that a≤τc and(106)lenc|θ,φθ≤δc.Note also thata is uniquely identified by φ.Proof.
We can rescale and reparameterizec to arc parameter, and we rescale and reparameterize c˜ at the same time (c˜ will not be by arc parameter in general); as discussed in Remarks 8, 10, and 11, the hypotheses and theses are unaffected by this action. Then we apply all previous results. Just note that(107)maxθ∈S1θ−φθ=maxθ∈S1θ−φ−1θ,for any diffeomorphism.Remark 12.
Actually, rerunning on the above proofs with some patience, we can improve the above thesis a bit. We add to the previous theorem these hypotheses: fix0<α≤1/2 and then 0<τ≤ατc/2 and suppose that we have c˜θ−cθ≤τ and(108)c˜′θ−c′θ≤αc′θ2,for all θ.
Then, all above thesis hold, moreover, there are two continuous functionsf,g:0,∞⟶ℝ (independent on α,τ) with f0=g0=1, such that(109)fα≤φ′≤gα.


#### 3.7.3. Asymmetry

Warning: the previous theorem seems symmetric, but it is not. The caveat is in the constantsτc,δc: it may be the case that they are quite different from τc˜,δc˜. In Figure 9, we see a piece of the two curves: curve c is blue and is flat; curve c˜ is red, and it has two inflections points A,B where the tangents are at an angle β which is as small as we would like, but then the inflection points can also be so close that the normals will cross before reaching curve c. So, while there is an easy way of representing c˜ using tubular coordinates around c, there is no way to find φ˜∈DiffS1,a˜ so as to write(110)cφθ=c˜θ+a˜θN˜θ.Figure 9
Curves for the example in Section3.7.3; curve c is blue and curve c˜ is red.


#### 3.7.4. Vice Versa

We have also a sort ofvice versa of the previous Theorem 3.Proposition 10 (derepresentation).
Supposec:S1⟶ℝ2 is a C2 immersed curve and c1,c2:S1⟶ℝ2 are given by tubular coordinates(111)ciθ=Φθ,aiθ=cθ+aiθNθ,where i=1,2 and a1,a2:S1⟶ℝ are of class C1; define(112)α=defa1−a2∞maxκ,β=def∂∂ca1−∂∂ca2∞,(where κ is the curvature of c). Then,(113)c1′θ−c2′θ≤c′θα2+β2,and (obviously) c1−c2∞=2ατc, for all θ.Proof.
We rescale and reparameterizec by arc parameter, and we rescale and reparameterize c1,c2 along with c, as explained in Remark 11; this operation is justified by Remarks 8 and 10; in particular, note that α,β are scale invariant; then,(114)ci′=T1−κai+ai′N,so(115)c1′−c2′=−Tκa1−a2+a1′−a2′N.
Hence,(116)c1′−c2′≤α2+β2.


#### 3.7.5. Loss of Regularity

Unfortunately, the representation discussed above suffers from a loss of regularity. Indeed, ifc:S1⟶ℝ2 is C2 and a:S1⟶ℝ is C∞, it may be the case that(117)c˜θ=Φθ,aθ=cθ+aθNθ,is of class C1 but not of class C2.This can be seen in very simple examples.Example 8.
Suppose that, fort near t=0, we have(118)ct=t,0,t<0,t,t33,t≥0,such a curve is C2 but not C3; then, for t≥0,(119)Tt=11+t41,t2,Nt=11+t4−t2,1,κ=2t1+t43/2,(that can be easily computed using a standard formula for curvature of planar curves, see Sec. 1.7.1 in [21]).
Choose thena≡1, so(120)c˜t=ct+Nt,but then(121)c˜t″=ct″+Nt″,and for t>0,(122)N″t=25t4−1t4+15/2,6t2t4−1t4+15/2.
So,(123)limt⟶0+c˜t″=−2,0,but c˜t″=0 for t<0.


#### 3.7.6. Neighborhoods of a Curve

The above results encode two different but equivalent ways to define a topology on the “manifold of immersed curves.” We specify them by describing the local bases of neighborhoods of a curvec.(i)
The “Banach way” in which a local base of open neighborhoods of a curvec is given by the sets(124)U=Uc,ε1=c˜:S1⟶ℝ2:c−c˜CR<ε1,whereε1>0 is small, and(125)cCR=defmaxθ∈S1cθ+c′θ+⋯+cR.(ii)
The “geometric way” in which a neighborhood in the local base is defined, forε2>0 small, as the set V=Vc,ε2 of all c˜ that can be expressed as in (88), namely,(126)c˜σ=Φφσ,aσ=cφσ+aσNφσ,for all choices ofa:S1⟶ℝ and ϕ∈DiffS1 with(127)aCR<ε2,φ−IdCR<ε2,where(128)φ−IdCR=maxθ∈S1dS1φθ,θ+φ′θ−1+⋯+φRθ,moreover derivatives of a may be computed in arc-parameter.The above are “equivalent” in this sense. Assume that the curvec is CR+1.(i)
For anyε1 that defines neighborhood U of the first type, there is small enough ε2 that defines a neighborhood V of the second type, so that V⊆U; this is easily proved (by using Leibnitz and Faa di Bruno formulas).(ii)
Consider now a neighborhoodV of the second type, for an ε2>0 small; for ε1 small enough, the previous results Theorem 3 and Remark 12 tells us that any curve c˜∈U can be expressed in tubular coordinates; since tubular coordinates are a local diffeomorphism, similar arguments as above (plus Theorem 2) show that (for ε1 even smaller) U⊆V.We skip details for sake of brevity.In all the above there is, however, an annoying condition: to prove equivalence ofCR neighborhoods, we have to assume that the curve c is CR+1. For this reason, this works well for defining topologies in the “manifold of smooth immersed curves”; in this case, we will use neighborhoods of the first kind (or, respectively, of the second kind) for all ε>0 and all R. This is the common approach, see [12].


#### 3.7.7. Local Injectivity

So far, we have consideredparametric curves. We have seen in Theorem 3 that we can represent nearby curves in a unique way using tubular coordinates, i.e., the map Φ.What happens when we considergeometric curves, that is, curves up to parameterization?Lemma 8 (local injectivity).
Letc be a C2 freely immersed planar curve. There exists a r=rc>0 such that, if(129)c˜s=defcs+asNs,c˜φs=cs+bsNs,with(130)a∞≤r,b∞≤r,∂a∂c∞≤12,∂b∂c∞≤12,where ∂/∂c is the arc derivation: then a≡b and φ is the identity.Proof.
If rescale the curvesc,c˜ and the functions a,b by a constant λ>0, and we rescale the constant rc by the same constant λ, then all hypotheses and theses are unaffected. So, we can assume with no loss of generality that c has length 2π.
If we reparameterizec^=c∘ψ then (cf the relation (93) in Remark 10) the functions a,b are reparameterized as well; having a^=a∘ψ, then ∂a^/∂c^=∂a/∂c∘ψ; and similarly for b, again hypotheses and theses are unaffected.
So, we can assume thatc is parameterized by arc parameter with no loss of generality.
Suppose that(131)c˜θ=cs+asNs,with a≤τc and a′≤1/2, then(132)c˜′=T1−κa+a′N.
So,(133)12≤c˜′≤94+14≤2.
Suppose moreover,(134)c˜φθ=cs+bsNs,with b′≤1/2, then(135)12≤c˜′φ′≤2,and then(136)14≤φ′≤4.
By contradiction, we may write(137)c˜nθ=cs+ansNs,c˜nφns=cs+bnsNs,with(138)an∞≤1n,bn∞≤1n,an′∞≤12,bn′∞≤12,where φn is not the identity: then, when 1/n<τc/4, the uniqueness condition (90) is contradicted, so there is a σn such that(139)lencn|σn,φnσn≥δc.
So, using Theorem2,(140)liminfn⟶∞dS1φn−1θn,θn>0.
Up to a subsequence, we can assume thatθn⟶θ˜ and(141)limn⟶∞dS1φn−1θn,θn=d˜>0.
We know that(142)14≤φn′≤4,up to a subsequence φn⟶φ uniformly and φn−1⟶φ−1 uniformly, where φ is a bi-Lip homeomorphism. Then, c˜n⟶c uniformly, and passing to the limit(143)c=c∘φ,so φ is a diffeomorphism. Moreover,(144)dS1φ−1θ˜,θ˜=d˜,so φ cannot be the identity; hence, c is not freely immersed.


#### 3.7.8. Auto Representation (This Section May Be Skipped on a First Read, Since It Is Not Needed in the following)

The above result is very important, but the proof gives no hint on what is going on. To this end, we drop the requirement that the curve be freely immersed and look at an easy question. Is it possible for a curve to represent itself locally?Lemma 9.
Letc be a C2 immersed planar curve. There are only finitely many ways in which the curve can represent itself geometrically, that is, finitely many choices a,ϕ(145)cs+asNs=cϕs,with ϕ∈DiffS1 and a:S1⟶ℝ continuous with a∞≤τc/2. In particular, there is a ρc>0 such that, if a∞≤ρc then a≡0.In particular, ifc is freely immersed, then a∞≤ρc implies a≡0 and ϕ≡Id.Proof.
Defineδc,τc as in Definition 17 (see also Proposition 7). Let Φs,t=cs+tNs. Consider the family P of all the pairs a,φ with a∞≤τc/2 and a≢0 and(146)Φs,as=cs+asNs=cφs,we will prove that there are only finitely many such pairs.
Hence, we will letPc be smaller than the minimum of a∞ for all such pairs:(147)Pc=12mina,φ∈Pa∞.
In the example, in Figure10, there are 3 pairs in P.
We have some very strong properties.(i)
Ifa1,φ1,a2,φ2∈P and there is a s˜∈S1s.t.φ1s˜=φ2s˜, then φ1≡φ2 and a1≡a2. Indeed, there is a small interval J containing s˜, where we can invert the map Φ and(148)s,ais=Φ−1cφis,that is, the first component ofΦ−1∘c is φi−1, so φ1≡φ2 in J; so, the set s∈S1:φ1s=φ2s is both open and closed. The previous argument also proves that a1≡a2.(ii)
Figure11 on the following page can be used as a visual guide in the following proof.LetI0⊂S1 be an open interval such that the length of |cI0 is less than δc and more than δc/2. As noted in Remark 9, δc≤lenc/2. Let U0 be the image of Φ for s∈I0 and t<τc. Choose ai,φi∈P with i∈1,2, let(149)Ii=φiI0,we will prove that eitherI1∩I2=∅, or I1=I2, φ1≡φ2, and a1≡a2. Assume that s˜∈I1∩I2, let si=φi−1s˜, then s1,s2∈I0; so,(150)cφisi=cs˜,using the relation (146),(151)Φsi,aisi=csi+aisiNs=cφisi=cs˜∈U,and the fact thatΦ is a diffeomorphism for s∈I0,t<τc and cs˜∈U, we obtain that s1=s2=s^ so φ1s^=φ2s^=s˜; hence, by the previous point φ1≡φ2, a1≡a2.(iii)
The differential ofΦ (computed using arc-length derivative) is(152)T1−κtds+Ndt,so the smallest principal value is at least 1/2. We can estimate the length of c for s∈Ii to be at least δc/4. Hence, there can be only finitely many such intervals.Figure 10
Example of curve with 3 pairs inP.Figure 11
Helper scheme for proof of Lemma9.Example 9.
The constantsρc in Lemma 9 and rc in Lemma 8 though cannot be estimated apriori by using differential quantities such as maxκ. These constants may arbitrarily smaller than the quantity τc (defined in Definition 17) that provides the width of the tubular neighborhood (Proposition 7). They really depend on how the curve is drawn.
This is seen in simple examples as follows:(i)
cz=εz+z2 for z∈S1⊂ℂ, or equivalently cθ=εcosθ+cos2θ,εsinθ+sin2θ for θ∈ℝ/2π∼S1that are small C∞ perturbations of the doubly traversed circle c2 seen in Example 1; see Figure 12. This curve is freely immersed, but it is quite near to the doubly traversed circle that is not freely immersed. For ε small, the curvature κc of c is approximatively 1; so, τc∼1/2, and(153)c2θ−cθ=ε,while(154)c′∼2±ε,c2′θ−c′θ=ε.
Hence,(155)c2′θ−c′θ∼ε2c′θ,so we can use Theorem 3 to express(156)c2θ=cθ+aθNθ,using tubular coordinates when ε<1/8, with a∼ε: so rc≤∼ε; similarly, we can represent cθ+π using c, so we have ρc≤∼ε.Figure 12
Curve in the Example9.


### 3.8. Free Immersions Are Open

We have thus come to a fundamental result.Theorem 4.
Free immersions are an open subset of immersions.We can detail and prove this fact in two ways.(i)
We can see it as a corollary of Lemma8. Consider a neighborhood V defined using the tubular coordinates; precisely, define V as in the second definition in Section 3.7.6, choosing ε2<minrc,1/2,τc/4 and R=2; knowing that c˜s≡c˜φs, we could choose a≡b, but then by Lemma 8, φ would be the identity. So, each and any curve in V is freely immersed.As discussed in Section3.7.6, this proves the result in the manifold of smooth immersed curves, where the above neighborhoods define a topology.(ii)
If we instead want to prove this for the standard BanachC2 topology (first definition in Section 3.7.6), we can proceed as follows. Suppose that cn is a sequence of immersed curves that are not free; and suppose that cn⟶c in C2. We may rescale and reparametrize all the curves so that all have length 2π and cn′≡1, and still cn⟶c in C2; we skip the details (see Theorem 7 in [20]); these assumptions simplify the following arguments. Let ϕn be a sequence such that cn=cn∘ϕn and ϕn is not the identity; as above, we know that ϕn′≡1 and we may choose ϕn to be a generator of the isotropy group, so that ϕnt=t+2π/jn; we know that jn is bounded by 4 times the curvature, so up to a subsequence jn is constant, let us call it j, and then ϕn=ϕ with ϕt=t+2π/j, and passing to the limits c∘ϕ=c, so c is not freely immersed.Note that both proofs need a compactness argument; this seems unavoidable since the size of the neighborhood cannot be estimated by using differentiable quantities, as explained in Example9.


## 3.1. Examples

Definition 15.
We start with some classical examples ofC∞ functions of compact support. Let(18)η0:ℝ⟶ℝ,η0u=defe4/π−1/u−1/π−u,u∈0,π,0,elsewhere,(19)η1:ℝ⟶ℝ,η1u=def42u−πe−1/u−1/π−u,u∈0,π,0,elsewhere,(see Figure 2 on the following page).Figure 2
Graph ofη0 as defined in (18); graph of η1 as defined in (19).We will use these to build some following examples.Example 3.
We present here a simple smooth formula for Example2(20)c:0,5π⟶ℝ2,cu=defsin2u,c2u,c2uη0u,u≤π,η0u−π,u≤2π,−η0u−2π,u≤3π,−η0u−3π,u≤4π,−η0u−4π,u≤5π.this is a C∞ function depicted at Figure 3 on the next page.Figure 3
x and y coordinates of curve defined in Example 2; trace of curve defined in Example 2.


### 3.1.1. Trace and Parameterization

If a curve is embedded then the curve is identified by its image, in these senses.(i)
Ifc,c˜:S1⟶ℝn are embedded and have the same image, then there is an unique reparameterization ϕ such that c˜=c∘ϕ.(ii)
Suppose thatc0:S1⟶ℝn is embedded and A=cS1 is the trace; suppose that c0 is parameterized by constant speed parameter; let us fix a candidate basepoint v∈A in the trace.We can state thatA,v characterize the embedded curve up to a choice of direction: precisely, there are exactly two different c1,c2:S1⟶ℝn, parameterized by constant speed parameter, such that c1θ0=c2θ0=v, and they satisfy(21)c1θ=c0θ+a,c2θ=c0b−θ,for unique choices of a,b∈S1 (dependant on v).In particular, if the rotation index ofc0 is r, then the latter curves have rotation indexes ±r.Since the definition offreely immersed curve says that the curve identifies a unique parameterization, then we may be induced to think that the above two properties extend to freely immersed curves: but this is not the case.Example 4.
The following two curves have the same trace, are freely immersed, are smooth, but have rotation indexes 0 and 1.(1)
This immersed closed curvec:0,2π⟶ℝ2 with components(22)c1u=sin2u,c2u=η12u,u∈0,π2,η12u−π2,u∈π2,π.(2)
This immersed closed curvec:0,2π⟶ℝ2 with components(23)c2u=sin2u,c2u=η12u,u∈0,π2,−η12u−π2,u∈π2,π.
See Figure4 on the following page.Figure 4
Trace of the two curves and parameterization defined in Example4; with color scheme used for the parameter.


## 3.1.1. Trace and Parameterization

If a curve is embedded then the curve is identified by its image, in these senses.(i)
Ifc,c˜:S1⟶ℝn are embedded and have the same image, then there is an unique reparameterization ϕ such that c˜=c∘ϕ.(ii)
Suppose thatc0:S1⟶ℝn is embedded and A=cS1 is the trace; suppose that c0 is parameterized by constant speed parameter; let us fix a candidate basepoint v∈A in the trace.We can state thatA,v characterize the embedded curve up to a choice of direction: precisely, there are exactly two different c1,c2:S1⟶ℝn, parameterized by constant speed parameter, such that c1θ0=c2θ0=v, and they satisfy(21)c1θ=c0θ+a,c2θ=c0b−θ,for unique choices of a,b∈S1 (dependant on v).In particular, if the rotation index ofc0 is r, then the latter curves have rotation indexes ±r.Since the definition offreely immersed curve says that the curve identifies a unique parameterization, then we may be induced to think that the above two properties extend to freely immersed curves: but this is not the case.Example 4.
The following two curves have the same trace, are freely immersed, are smooth, but have rotation indexes 0 and 1.(1)
This immersed closed curvec:0,2π⟶ℝ2 with components(22)c1u=sin2u,c2u=η12u,u∈0,π2,η12u−π2,u∈π2,π.(2)
This immersed closed curvec:0,2π⟶ℝ2 with components(23)c2u=sin2u,c2u=η12u,u∈0,π2,−η12u−π2,u∈π2,π.
See Figure4 on the following page.Figure 4
Trace of the two curves and parameterization defined in Example4; with color scheme used for the parameter.


## 3.2. Reparameterizations and Isotropy Group

Lemma 2.
Ifϕ∈Diff−S1, then ϕ has two fixed points.Proof.
We representϕ as a map ϕ:0,2π⟶ℝ that is continuous, strictly decreasing and such that(24)ϕ0∈2π,4π,ϕ2π=ϕ0−2π,then(25)ϕ2π∈0,2π,so the graph y=ϕx must intersect both the graph y=x and the graph y=x+2π for two different points x1,x2∈0,2π that are the two fixed points.Lemma 3.
Ifc is immersed and ψ∈Gc, then ψ∈Diff+S1.Proof.
Suppose thatψ∈Diff−S1, let u∈S1 be a fixed point (by Lemma 2). By deriving(26)c′θ=c′ψθψ′θ,setting θ=u(27)c′u=c′uψ′u,and this is impossible since ψ′u<0.


## 3.3. Local Embedding

### 3.3.1. Length of Curve Arcs

Definition 16.
Supposec:S1⟶ℝn is C1. Let σ,σ˜∈S1. When σ≠σ˜ there are two arcs in S1 connecting σ to σ˜. By(28)lenc|σ,σ˜,we will mean the minimum of the lengths of c when restricted to one of the two arcs connecting σ to σ˜.Ifc is periodically extended to c:ℝ⟶ℝn and σ,σ˜∈ℝ, then there is an unique k∈ℤ such that(29)σ≤σ˜+k2π<σ+2π,and then, letting(30)l1=∫σσ˜+k2πc′θdθ,l2=∫σ˜+k2πσ+2πc′θdθ,we define(31)lenc|σ,σ˜=minl1,l2.In particular, whenc is parameterized at constant speed (i.e.,c′≡ℓ), then we will (covertly) assume that σandσ˜ are chosen (up to adding k2π) so that σ−σ˜=dS1σ,σ˜≤π and then(32)lenc|σ,σ˜=ℓσ−σ˜.Remark 7.
Whenc is not parameterized by constant velocity, the above may lead to some confusion. The interval σ,σ˜ in the notation (28) implicitly refers to the choice of arc in S1 that provides the above minimum. Note that this may not be the shortest arc connecting σ to σ˜ in S1. This may happen if the parameterization of c has regions of fast and slow velocity, as in this example.Example 5.
Letc˜θ=cosθ,sinθ be the standard circle, and(33)ϕθ=4θ,0≤θ≤π3,25x+3π,π3≤θ≤2π,(see plot in Figure 5) then smooth out the corners of ϕ so that it becomes a diffeomorphism of S1; let(34)c=c˜∘ϕ,let θ0=0,θ1=π/3 in S1∼ℝ/2π, then dS1θ0,θ1=π/3, and is given by the arc moving counterclockwise from θ0 to θ1, while(35)lenc|σ0,σ1=2π3,is given by the arc moving clockwise from θ0 to θ1.Figure 5
Plot of functionϕ in equation (33) in Example 5.This never happens for small distances/lengths, though.Theorem 2.
Fix an immersed curvec:S1⟶ℝn; let(36)M=maxc′,m=minc′,σc=def1M/m+1.(i)
For anyθ0,θ1 in S1 such that(37)dS1θ0,θ1<2πσc,the shortest arc connecting them inS1 is also the arc where(38)lenc|θ0,θ1is computed(ii)
For anyθ0,θ1inS1such that(39)lenc|θ0,θ1<σclencthe arc where(40)lenc|θ0,θ1,is computed is also the shortest arc connecting them inS1, whose length is(41)dS1θ0,θ1.(iii)
In any of the above cases,(42)mdS1θ0,θ1≤lenc|θ0,θ1≤MdS1θ0,θ1.


### 3.3.2. Estimates

We begin with this estimate.Proposition 5.
Letαs be the angle function, for s∈J=0,2π. The fact that the curve is closed imposes lower bounds on maxJα′.(i)
If the rotation indexℐ of the curve is not zero, then α2π−α0=2πI so necessarily maxJα′≥I(ii)
If the rotation index of the curve is zero, then necessarilymaxJα′≥1/2
Indeed, we can prove that(43)maxJα−minJα≥π;otherwise, let(44)β=maxJα+minJα2,taking v=cosβ,sinβ, we would have(45)v⋅c′θ=c′θcosβ−αθ>0,for all θ, hence the curve would not be closed.Definition 17.
Givenc:S1⟶ℝ2, a C2 immersed closed curve, we recall that κ is the scalar curvature of c; we define (46)δc=defπ3maxκ,τc=def12maxκ.Note that since the curve is closed, κ cannot be identically zero.Note thatδc=τc2π/3, but we define two quantities since this simplifies the notation in the following. We have 2τc≤δc but 3τc≥δc.Remark 8.
Note that if we rescale the curvec by a factor λ, then δc,τc and lenc|σ,σ˜ are multiplied by λ as well. If we rotate or translate c, then δc,τc and lenc|σ,σ˜ are unaffected. If we reparameterize, then δc,τc are unchanged, whereas if ψ∈DiffS1 and c˜=c∘ψ, we have(47)lenc˜|ψσ,ψσ˜=lenc|σ,σ˜.
In all (with the exception of relation (100) in Lemma 7) following definitions, propositions, and theorems, the formulas are built to be “geometrical”: this means that, if the curves are reparameterized, rescaled, translated or rotated, then the formulas change in predictable ways (as explained above).
This simplifies the proofs: in the proofs we can assume, with no loss of generality, that the curve is parameterized by arc parameter.Remark 9.
Note thatδc≤lenc/3 for curves of index zero and δc≤lenc/6I for curves of index I≠0.Proof.
We use Proposition5. The formula in the thesis is invariant for reparameterizations and scaling; we rescale the curve so that lenc=2π and reparameterize by arc parameter so that c′≡1. For curves of index zero, the thesis π/3maxJκ≤lenc/3, that is, π≤maxJκlenc; since c′≡1 by (16), this last becomes 1/2≤maxJα′ that was proved above. For curves of index I, the thesis π/3maxJκ≤lenc/6I, that is, 2πI≤maxJκlenc, then becomes I≤maxJα′ that was proved above.ForI≠0 the above is sharp, as in the case of cθ=cosIθ,sinIθ.


### 3.3.3. Local Embedding of Curves

It is well known that aC1 immersion ℐ:M⟶N is a local embedding. For curves of class C2, we can provide a simple quantitative statement.Proposition 6 (local embedding).
Letc:S1⟶ℝ2 a C2 immersed curve. Define δc as in Definition 17. For any a,b∈S1, let L=deflenc|a,b; assume that L≤2δc, then cb−ca≥L/2; so, c|a,b is embedded.Proof.
For simplicity, we assume thatc is periodically extended to c:ℝ⟶ℝ2; then, we identify the interval in ℝ that is associated to the arc of the curve where the length lenc|a,b is computed; for simplicity, we call this interval a,b again. (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
Using Lemma1 and Remark 8, assume that c′≡ℓ=lenc/2π; then, ∂s=1/ℓ∂θ, so T=c′/ℓ and(48)lenc|a,b=ℓb−a.
As noted in (16)(49)κ=∂sT=T′ℓ=α′ℓ,so δc=ℓπ/3maxα′. Let m=a+b/2 be the middle point. Let αt be the angle function (14). Up to rotation, suppose c′m=ℓ,0,Tm=1,0; so, we can assume αm=0. Let L=lenc|a,b=ℓb−a≤2δc. For any θ∈a,b, we have ℓθ−m≤δc; hence, θ−m≤π/3maxα′; hence, for all a≤θ≤b, we have(50)αθ≤θ−mmaxα′≤π3,and hence, cosαθ≥1/2; hence, for a≤θ1≤θ2≤b, for the abscissa, we can write(51)c1θ2−c1θ1=ℓ∫θ1θ2cosαθdθ≥ℓθ2−θ112.


## 3.3.1. Length of Curve Arcs

Definition 16.
Supposec:S1⟶ℝn is C1. Let σ,σ˜∈S1. When σ≠σ˜ there are two arcs in S1 connecting σ to σ˜. By(28)lenc|σ,σ˜,we will mean the minimum of the lengths of c when restricted to one of the two arcs connecting σ to σ˜.Ifc is periodically extended to c:ℝ⟶ℝn and σ,σ˜∈ℝ, then there is an unique k∈ℤ such that(29)σ≤σ˜+k2π<σ+2π,and then, letting(30)l1=∫σσ˜+k2πc′θdθ,l2=∫σ˜+k2πσ+2πc′θdθ,we define(31)lenc|σ,σ˜=minl1,l2.In particular, whenc is parameterized at constant speed (i.e.,c′≡ℓ), then we will (covertly) assume that σandσ˜ are chosen (up to adding k2π) so that σ−σ˜=dS1σ,σ˜≤π and then(32)lenc|σ,σ˜=ℓσ−σ˜.Remark 7.
Whenc is not parameterized by constant velocity, the above may lead to some confusion. The interval σ,σ˜ in the notation (28) implicitly refers to the choice of arc in S1 that provides the above minimum. Note that this may not be the shortest arc connecting σ to σ˜ in S1. This may happen if the parameterization of c has regions of fast and slow velocity, as in this example.Example 5.
Letc˜θ=cosθ,sinθ be the standard circle, and(33)ϕθ=4θ,0≤θ≤π3,25x+3π,π3≤θ≤2π,(see plot in Figure 5) then smooth out the corners of ϕ so that it becomes a diffeomorphism of S1; let(34)c=c˜∘ϕ,let θ0=0,θ1=π/3 in S1∼ℝ/2π, then dS1θ0,θ1=π/3, and is given by the arc moving counterclockwise from θ0 to θ1, while(35)lenc|σ0,σ1=2π3,is given by the arc moving clockwise from θ0 to θ1.Figure 5
Plot of functionϕ in equation (33) in Example 5.This never happens for small distances/lengths, though.Theorem 2.
Fix an immersed curvec:S1⟶ℝn; let(36)M=maxc′,m=minc′,σc=def1M/m+1.(i)
For anyθ0,θ1 in S1 such that(37)dS1θ0,θ1<2πσc,the shortest arc connecting them inS1 is also the arc where(38)lenc|θ0,θ1is computed(ii)
For anyθ0,θ1inS1such that(39)lenc|θ0,θ1<σclencthe arc where(40)lenc|θ0,θ1,is computed is also the shortest arc connecting them inS1, whose length is(41)dS1θ0,θ1.(iii)
In any of the above cases,(42)mdS1θ0,θ1≤lenc|θ0,θ1≤MdS1θ0,θ1.


## 3.3.2. Estimates

We begin with this estimate.Proposition 5.
Letαs be the angle function, for s∈J=0,2π. The fact that the curve is closed imposes lower bounds on maxJα′.(i)
If the rotation indexℐ of the curve is not zero, then α2π−α0=2πI so necessarily maxJα′≥I(ii)
If the rotation index of the curve is zero, then necessarilymaxJα′≥1/2
Indeed, we can prove that(43)maxJα−minJα≥π;otherwise, let(44)β=maxJα+minJα2,taking v=cosβ,sinβ, we would have(45)v⋅c′θ=c′θcosβ−αθ>0,for all θ, hence the curve would not be closed.Definition 17.
Givenc:S1⟶ℝ2, a C2 immersed closed curve, we recall that κ is the scalar curvature of c; we define (46)δc=defπ3maxκ,τc=def12maxκ.Note that since the curve is closed, κ cannot be identically zero.Note thatδc=τc2π/3, but we define two quantities since this simplifies the notation in the following. We have 2τc≤δc but 3τc≥δc.Remark 8.
Note that if we rescale the curvec by a factor λ, then δc,τc and lenc|σ,σ˜ are multiplied by λ as well. If we rotate or translate c, then δc,τc and lenc|σ,σ˜ are unaffected. If we reparameterize, then δc,τc are unchanged, whereas if ψ∈DiffS1 and c˜=c∘ψ, we have(47)lenc˜|ψσ,ψσ˜=lenc|σ,σ˜.
In all (with the exception of relation (100) in Lemma 7) following definitions, propositions, and theorems, the formulas are built to be “geometrical”: this means that, if the curves are reparameterized, rescaled, translated or rotated, then the formulas change in predictable ways (as explained above).
This simplifies the proofs: in the proofs we can assume, with no loss of generality, that the curve is parameterized by arc parameter.Remark 9.
Note thatδc≤lenc/3 for curves of index zero and δc≤lenc/6I for curves of index I≠0.Proof.
We use Proposition5. The formula in the thesis is invariant for reparameterizations and scaling; we rescale the curve so that lenc=2π and reparameterize by arc parameter so that c′≡1. For curves of index zero, the thesis π/3maxJκ≤lenc/3, that is, π≤maxJκlenc; since c′≡1 by (16), this last becomes 1/2≤maxJα′ that was proved above. For curves of index I, the thesis π/3maxJκ≤lenc/6I, that is, 2πI≤maxJκlenc, then becomes I≤maxJα′ that was proved above.ForI≠0 the above is sharp, as in the case of cθ=cosIθ,sinIθ.


## 3.3.3. Local Embedding of Curves

It is well known that aC1 immersion ℐ:M⟶N is a local embedding. For curves of class C2, we can provide a simple quantitative statement.Proposition 6 (local embedding).
Letc:S1⟶ℝ2 a C2 immersed curve. Define δc as in Definition 17. For any a,b∈S1, let L=deflenc|a,b; assume that L≤2δc, then cb−ca≥L/2; so, c|a,b is embedded.Proof.
For simplicity, we assume thatc is periodically extended to c:ℝ⟶ℝ2; then, we identify the interval in ℝ that is associated to the arc of the curve where the length lenc|a,b is computed; for simplicity, we call this interval a,b again. (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
Using Lemma1 and Remark 8, assume that c′≡ℓ=lenc/2π; then, ∂s=1/ℓ∂θ, so T=c′/ℓ and(48)lenc|a,b=ℓb−a.
As noted in (16)(49)κ=∂sT=T′ℓ=α′ℓ,so δc=ℓπ/3maxα′. Let m=a+b/2 be the middle point. Let αt be the angle function (14). Up to rotation, suppose c′m=ℓ,0,Tm=1,0; so, we can assume αm=0. Let L=lenc|a,b=ℓb−a≤2δc. For any θ∈a,b, we have ℓθ−m≤δc; hence, θ−m≤π/3maxα′; hence, for all a≤θ≤b, we have(50)αθ≤θ−mmaxα′≤π3,and hence, cosαθ≥1/2; hence, for a≤θ1≤θ2≤b, for the abscissa, we can write(51)c1θ2−c1θ1=ℓ∫θ1θ2cosαθdθ≥ℓθ2−θ112.


## 3.4. Isotropy Group Is Discrete

Given an immersionℐ:M⟶N, it is possible to prove that the isotropy group is discrete (when M is paracompact) and even finite (when M is compact; this latter result appears in [5]). When considering curves, we can obtain the same results (and even more) in a more direct and geometric way.Lemma 4.
Letc:S1⟶ℝn be immersed.(i)
Gcis finite(ii)
Ifc∘ψ≡c∘ϕandψa=ϕafor ana∈S1, thenψ≡ϕ(iii)
Ifc is parameterized by constant speed (see Lemma 1), then there is a k∈ℕ,k≥1 s.t. Gc is the set of all ϕt=t+2πj/k for j=0,…,k−1 (the proof of this is a special case of the 2nd step of the proof of Lemma 9)Proof.
(i)
We prove the third point. Indeed derivingc=c∘ϕ and noting that c′≡ℓ, we obtain ϕ′≡1 so ϕt=t+β; hence, ϕjt=t+jβ∈Gc, for all j; if β/π is irrational, then jβ would be dense in S1=ℝ/2π, and this is denied by Proposition 6. Moreover, if ϕt=t+2π/k and c=c∘ϕ, then c0=c2π/k; but by Proposition 6, 2π/3maxκ≤2πℓ/k, that is, k≤3ℓmaxκ. So, there is an unique k such that any ϕ∈Gc can be written as ϕt=t+2πj/k.(ii)
The above characterization shows that ifϕ′≡1≡ψ′ and ϕa=ψa, then ϕ≡ψ. By Remark 2, this is valid for any curve (even when it is not parameterized by constant speed). This proves the second point.(iii)
The first point follows again from Remark2.


## 3.5. Tubular Neighborhoods

Existence of tubular neighborhood is well known; we provide a quantitative result forC2 planar immersed curves.Proposition 7 (tubular neighborhood).
Defineδc,τc as in Definition 17. Fix a,b∈S1 with lenc|a,b≤2δc. Let(52)Φ:a,b×−τc,τc⟶ℝ2,Φs,t=cs+tNs,then Φ is a diffeomorphism with its image. Moreover, if the arc s1,s2 is contained in the arc a,b identified above, then(53)Φs1,t1−Φs2,t2≥14lenc|s1,s2,whereas (obviously)(54)Φs,t1−Φs,t2=t2−t1.Proof.
Assume that the curve has length2π and is parameterized in arc parameter; with no loss of generality (recalling Remark 8); let α be the angle function (14).
Extendc to a periodic function c:ℝ⟶ℝ2 and identify the interval in ℝ that is associated with the arc of the curve where the length lenc|a,b is computed. For simplicity, we call this interval a,b again (if the arc is short enough, then by Theorem 2, no ambiguity is possible).
The Jacobian ofΦ is(55)∂∂sΦ=T1−κt,∂∂tΦ=N,so its determinant is 1−κt≥1/2 by the hypothesis t≤τc.
We will then prove thatΦ is injective, so it will be an homeomorphism with its image, and since it is a local diffeomorphism, it will be a diffeomorphism.
Chooses1,t1 and s2,t2 with a≤s1<s2≤b and t1≤τc, t2≤τc.
We setm=s2+s1/2. Up to rotation, we assume that Tm=e1=1,0 and αm=0, so that Nm is perpendicular to the x axis. As in Proposition 6, we can prove that cosαs≥1/2 for all s1≤s≤s2.
We write(56)Φs,t=cs+tNs=cm+∫msTsdθ+tNs,and then for the abscissa(57)Φs,t1=cm1+∫mscosαθdθ+tsinαs,note that Φm,t1=cm1. Deriving, we obtain(58)∂∂sΦs,t1=cosαs1−tα′s≥14.
We then obtain that(59)Φs2,t21−Φm,t21≥14s2−m,while(60)Φm,t11−Φs1,t11≥14m−s1,and recalling that Φm,t11=Φm,t21=cm1 and summing, we obtain(61)Φs2,t21−Φs1,t11≥14s2−s1.We will call tubular coordinates aroundc the formula (52).


### 3.5.1. Counterexample

The hypothesis “c∈C2” in the previous proposition may be broadened to “c∈C1,1”; but the results fails if we only assume that “c∈C1,α” with α∈0,1, as seen in this example (adapted from [1]).Example 6.
Let0<α<1 and cθ=θ,θ1+α; then, for θ>0,(62)cθ+aNθ=θ−aα+1θαα+12θ2α+1,aα+12θ2α+1+θα+1,and this meets the y axes for(63)a=a˜θ=θ1−αα+12θ2α+1α+1.
So, by symmetry,(64)cθ+a˜θNθ=c−θ+a˜−θN−θ,and at the same time,(65)limθ⟶0a˜θ=0.


### 3.5.2. Nearby Points

Supposec:S1⟶ℝ2 is a C2 immersed curve. Let 0<τ≤τc and Vτ=S1×−τ,τ and(66)Φ:Vτ⟶ℝ2,Φs,t=cs+tNs,and Uτ=ΦVτ.Proposition 8.
Uτ is also the set of points at distance at most τ from the trace cS1.Proof.
LetK=cS1 be the trace of the curve (it is a compact subset of ℝ2). We use the distance function dK:ℝ2⟶ℝ defined as(67)dKx≔infy∈Kx−y,(for an introduction to this object, see [14] and references therein).
Letx∈Uτ, there is a θ,t∈Vτ such that(68)x=cθ+tNθ.
So,(69)x−cθ≤t≤τ,then let θ˜ be a minimum for(70)x−cθ.
So, clearly,(71)dKx=x−cθ˜≤τ.
Vice versa ifdKx≤τ let θ˜ be a minimum as above, then geometrical considerations tell that the segment from x to cθ˜ is orthogonal to the tangent Tθ˜ at cθ˜.As a corollary of Proposition7, for any such neighborhood of the image of c, the “projection to c” is a C1 multivalued map (with finitely many projections in S1).


## 3.5.1. Counterexample

The hypothesis “c∈C2” in the previous proposition may be broadened to “c∈C1,1”; but the results fails if we only assume that “c∈C1,α” with α∈0,1, as seen in this example (adapted from [1]).Example 6.
Let0<α<1 and cθ=θ,θ1+α; then, for θ>0,(62)cθ+aNθ=θ−aα+1θαα+12θ2α+1,aα+12θ2α+1+θα+1,and this meets the y axes for(63)a=a˜θ=θ1−αα+12θ2α+1α+1.
So, by symmetry,(64)cθ+a˜θNθ=c−θ+a˜−θN−θ,and at the same time,(65)limθ⟶0a˜θ=0.


## 3.5.2. Nearby Points

Supposec:S1⟶ℝ2 is a C2 immersed curve. Let 0<τ≤τc and Vτ=S1×−τ,τ and(66)Φ:Vτ⟶ℝ2,Φs,t=cs+tNs,and Uτ=ΦVτ.Proposition 8.
Uτ is also the set of points at distance at most τ from the trace cS1.Proof.
LetK=cS1 be the trace of the curve (it is a compact subset of ℝ2). We use the distance function dK:ℝ2⟶ℝ defined as(67)dKx≔infy∈Kx−y,(for an introduction to this object, see [14] and references therein).
Letx∈Uτ, there is a θ,t∈Vτ such that(68)x=cθ+tNθ.
So,(69)x−cθ≤t≤τ,then let θ˜ be a minimum for(70)x−cθ.
So, clearly,(71)dKx=x−cθ˜≤τ.
Vice versa ifdKx≤τ let θ˜ be a minimum as above, then geometrical considerations tell that the segment from x to cθ˜ is orthogonal to the tangent Tθ˜ at cθ˜.As a corollary of Proposition7, for any such neighborhood of the image of c, the “projection to c” is a C1 multivalued map (with finitely many projections in S1).


## 3.6. Not a Covering Map

By looking at the previous Proposition7, we may think that Φ is the universal covering map of U=ΦV (see [17] for the definition). This would be very convenient, and indeed, we could use the lifting lemma to ease some of the following proofs.Supposec˜,c:S1⟶ℝ2 are C2 immersed curves. Consider this statement, that is usually called lifting lemma:If the trace ofc˜ is contained in Uτ, then there is a choice of continuous φ:ℝ⟶ℝ, a:ℝ⟶−τ,τ such that(72)c˜θ=Φφθ,aθ=cφθ+aθNφθ,∀θ.Unfortunately, this is not the case, as seen in this example in Figure6, where the curve c is blue and the curve c˜ is red. The trace of the curve c˜ is all contained in the open set Uτc, but representation (72) cannot hold. We can though prove a version of the lifting lemma useful in the following.Figure 6
Figure for example in Section3.6; the curve c is blue and the curve c˜ is red.


## 3.7. Neighborhoods

### 3.7.1. Nearby Projection

Lemma 5 (nearby projection).
Fix aCR immersed curve c, with R≥2.(1)
Ifx∈ℝ2 and σ˜∈S1 and(73)d=defx−cσ˜<δc4,then there is ana∈ℝ with a≤d and a σ∈S1 with(74)lenc|σ,σ˜≤4d,such that(75)x=cσ+aNσ.Note also thata is uniquely identified by σ.(2)
They are unique in the family ofσ,a such that a≤τc and(76)lenc|σ,σ˜≤δc,so we can seeσ,a as functions of x, as follows.(3)
Considerx˜∈ℝ2 and σ˜∈S1 for which(77)d=defx˜−cσ˜≤τc2,let ε>0 small such that d+ε<δc/4 and let B=Bx˜,ε=x∈ℝ2:x−x˜<ε for convenience. There is a choice of function a,φ:B⟶ℝ of class CR−1 such that(78)x=Φφx,ax=cφx+axNφx,for all x∈B, and they are unique as specified above.
Note thatτc/2<δc/4<τc.Proof.
Supposec is by arc parameter (with no loss of generality as explained in Remark 8); so we write (recalling (32)) a−b instead of lenc|a,b.(i)
Choosex,σ˜ as in the statement and let Jσ˜=σ˜−δc,σ˜+δc, then consider any minimum point θ^ of(79)minθ∈Jσ˜x−cθ,note that the minimum value has to be less thand; so,(80)cσ˜−cθ^≤cσ˜−x+x−cθ^≤2d,but at the same time (sinceσ˜ and θ^ are at arc distance at most δc), by the previous Proposition 6.(81)cσ˜−cθ^≥12θ^−σ˜.So, combining the two(82)4d≥θ^−σ˜,but4d<δc so θ^ is not at extremes. Then, any θ^ providing the minimum must be internal in the interval Jσ˜: by geometrical reasoning the segment from x to cθ^ is orthogonal to the curve so there is a a such that(83)x=Φθ^,a.(ii)
Recall thatδc/4≤τc; the map Φ is injective for θ∈Jσ˜ and a≤τc, so θ^,a are unique.(iii)
For anyx∈B, we have(84)x−cσ˜<d+ε,and since d+ε<δc/4, then there is an unique σ∈Jσ˜ and a with a≤τc such that(85)x=cσ+aNσ,and we denote them by σ=φx,a=ax. Moreover, we can invert the function(86)Φ:Jσ˜×−τc,τc⟶ℝ2,and write(87)φx,ax=Φ−1x,for x∈B. This proves that φ,a∈C1B.


### 3.7.2. Global Lifting

Proposition 9 (global lifting).
Supposec:S1⟶ℝ2 is a CR immersed curve and c˜:S1⟶ℝ2 is CR−1, with R≥2. Fix 0<τ<δc/4. Suppose that we have c˜θ−cθ≤τ for all θ. There exists choice of a:S1⟶ℝ and φ:S1⟶S1 such that(88)c˜σ=Φφσ,aσ=cφσ+aσNcφσ,∀σ∈S1,with aσ≤τ and(89)lenc|σ,φσ≤4τ,holding for all σ. And, they are unique in the class of CR−1 functions such that a≤τc and(90)lenc|σ,φσ≤δc.Note also thata is uniquely identified by φ.Proof.
We just substitutex=c˜σ in the previous Lemma. By the second point, we can define functions φθandaθ uniquely as prescribed. By the third point, they are C1.Remark 10.
Suppose we are given two curvesc1,c2 and we know that there exists a choice a,φ such that(91)c2σ=c1φσ+aσNc1φσ,∀σ∈S1,as in (88). If we rotate or translate the two curves, then the above relation will hold, with the same a,φ. If we rescale the two curves by λ>0, then the relation will hold with λa,φ.
If we chooseψ∈Diff+S1 and we reparameterize all curves at the same time by c˜i=ci∘ψ, then(92)c˜2s=c˜1φ˜s+a˜sNc˜1φ˜s,∀s∈S1,holds for(93)a˜=a∘ψ,φ˜=ψ−1∘φ∘ψ.
This follows from direct computation and Remark5.Example 7.
So far so good, butφ may fail to be a diffeomorphism, as in this simple example in Figure 7. where the curve c is blue and the curve c˜ is red.But some simple lemmas can help.Figure 7
Curves in Example7; curve c is blue and curve c˜ is red.Figure 8
Scheme for Lemma 6.Lemma 6.
Let0<α<1. If we have w,v∈ℝn such that(94)w−v≤αv,then the angle β between v and w satisfies β≤arcsinα, and moreover,(95)1−αv≤v⋅w≤1+αv,1−αv≤w≤1+αv.See Figure 8 on the next page.Remark 11.
Letα>0. Suppose c1,c2,c3:S1⟶ℝ2 are C1 maps and(96)c2′θ−c3′θ≤αc1′θ,if ψ∈Diff+S1 and we reparameterize all curves at the same time by c˜i=ci∘ψ, then(97)c˜2′θ−c˜3′θ≤αc˜3′θ.
Similarly, if we rescale, rotate, or translate all curves at the same time.So, (96) is a geometric estimate, indeed we may rewrite it as(98)∂c1c2θ−∂c1c3θ≤α,where ∂c1=1/c1′∂/∂θ.Lemma 7.
Assume all hypotheses in Proposition 9. Assume moreover thatτ≤τc/4 and assume that(99)c˜′σ−c′σ≤12c′σ,∀σ∈S1,then c˜ is immersed and φ is a diffeomorphism. Moreover, when c is parameterized at constant speed, we can state that(100)15≤φ′≤4.Proof.
We rescale and reparameterizec to arc parameter using a reparameterization ψ, and at the same time, we rescale and reparameterize c˜ using the same rescaling and ψ (note that c˜ is not necessarily by arc parameter); with no loss in generality, as explained in Remarks 10 and 11.
Letβ be the angle between c˜′σ and c′σ: by Lemma 6, β=arcsinα, so it is at most π/6. Let θ=φσ; we know that σ−θ≤4τ≤τc; the angle γ between c′σ and c′θ is at most σ−θmaxk so γ≤1/2: so the angle β+γ between c˜′σ and c˜′φσ is at most 1/2+π/6, and this is less than π/2.
Deriving inσ and assuming that c is by arc parameter(101)c˜′σ=Tφ′σ1−κa+a′N,where T,N,κ are evaluated at φσ; then,(102)c˜′⋅T=φ′σ1−κa,now if a≤τc then 1−κa≥1/2; moreover, by the above reasoning c˜′⋅T>0, so φ′σ>0. Moreover, we note that 1/2≤c˜′≤3/2, 1/2≤1−κa≤3/2 and cos1/2+π/6≤6/10 to prove (100). Relation (93) tells then that φ will always be a diffeomorphism, for any curve satisfying the hypotheses.We summarize all the above: we show sufficient hypothesis such thatc˜ may be represented intubular coordinates aroundc.Theorem 3 (representation theorem).
Supposec:S1⟶ℝ2 is a CR immersed curve and c˜:S1⟶ℝ2 is CR−1, with R≥2. Define δc,τc as in Definition 17. Fix 0<τ≤τc/4. Suppose that we have c˜θ−cθ≤τ and(103)c˜′θ−c′θ≤c′θ2,for all θ.
Then,c˜ is immersed, there are φ∈Diff+S1 and a:S1⟶−τ,τ of class CR−1 such that(104)c˜φθ=Φθ,aθ=cθ+aθNθ,∀θ,with aσ≤τ and(105)lenc|σ,φσ≤4τ,holding for all σ.
They are unique in the class ofC1 functions such that a≤τc and(106)lenc|θ,φθ≤δc.Note also thata is uniquely identified by φ.Proof.
We can rescale and reparameterizec to arc parameter, and we rescale and reparameterize c˜ at the same time (c˜ will not be by arc parameter in general); as discussed in Remarks 8, 10, and 11, the hypotheses and theses are unaffected by this action. Then we apply all previous results. Just note that(107)maxθ∈S1θ−φθ=maxθ∈S1θ−φ−1θ,for any diffeomorphism.Remark 12.
Actually, rerunning on the above proofs with some patience, we can improve the above thesis a bit. We add to the previous theorem these hypotheses: fix0<α≤1/2 and then 0<τ≤ατc/2 and suppose that we have c˜θ−cθ≤τ and(108)c˜′θ−c′θ≤αc′θ2,for all θ.
Then, all above thesis hold, moreover, there are two continuous functionsf,g:0,∞⟶ℝ (independent on α,τ) with f0=g0=1, such that(109)fα≤φ′≤gα.


### 3.7.3. Asymmetry

Warning: the previous theorem seems symmetric, but it is not. The caveat is in the constantsτc,δc: it may be the case that they are quite different from τc˜,δc˜. In Figure 9, we see a piece of the two curves: curve c is blue and is flat; curve c˜ is red, and it has two inflections points A,B where the tangents are at an angle β which is as small as we would like, but then the inflection points can also be so close that the normals will cross before reaching curve c. So, while there is an easy way of representing c˜ using tubular coordinates around c, there is no way to find φ˜∈DiffS1,a˜ so as to write(110)cφθ=c˜θ+a˜θN˜θ.Figure 9
Curves for the example in Section3.7.3; curve c is blue and curve c˜ is red.


### 3.7.4. Vice Versa

We have also a sort ofvice versa of the previous Theorem 3.Proposition 10 (derepresentation).
Supposec:S1⟶ℝ2 is a C2 immersed curve and c1,c2:S1⟶ℝ2 are given by tubular coordinates(111)ciθ=Φθ,aiθ=cθ+aiθNθ,where i=1,2 and a1,a2:S1⟶ℝ are of class C1; define(112)α=defa1−a2∞maxκ,β=def∂∂ca1−∂∂ca2∞,(where κ is the curvature of c). Then,(113)c1′θ−c2′θ≤c′θα2+β2,and (obviously) c1−c2∞=2ατc, for all θ.Proof.
We rescale and reparameterizec by arc parameter, and we rescale and reparameterize c1,c2 along with c, as explained in Remark 11; this operation is justified by Remarks 8 and 10; in particular, note that α,β are scale invariant; then,(114)ci′=T1−κai+ai′N,so(115)c1′−c2′=−Tκa1−a2+a1′−a2′N.
Hence,(116)c1′−c2′≤α2+β2.


### 3.7.5. Loss of Regularity

Unfortunately, the representation discussed above suffers from a loss of regularity. Indeed, ifc:S1⟶ℝ2 is C2 and a:S1⟶ℝ is C∞, it may be the case that(117)c˜θ=Φθ,aθ=cθ+aθNθ,is of class C1 but not of class C2.This can be seen in very simple examples.Example 8.
Suppose that, fort near t=0, we have(118)ct=t,0,t<0,t,t33,t≥0,such a curve is C2 but not C3; then, for t≥0,(119)Tt=11+t41,t2,Nt=11+t4−t2,1,κ=2t1+t43/2,(that can be easily computed using a standard formula for curvature of planar curves, see Sec. 1.7.1 in [21]).
Choose thena≡1, so(120)c˜t=ct+Nt,but then(121)c˜t″=ct″+Nt″,and for t>0,(122)N″t=25t4−1t4+15/2,6t2t4−1t4+15/2.
So,(123)limt⟶0+c˜t″=−2,0,but c˜t″=0 for t<0.


### 3.7.6. Neighborhoods of a Curve

The above results encode two different but equivalent ways to define a topology on the “manifold of immersed curves.” We specify them by describing the local bases of neighborhoods of a curvec.(i)
The “Banach way” in which a local base of open neighborhoods of a curvec is given by the sets(124)U=Uc,ε1=c˜:S1⟶ℝ2:c−c˜CR<ε1,whereε1>0 is small, and(125)cCR=defmaxθ∈S1cθ+c′θ+⋯+cR.(ii)
The “geometric way” in which a neighborhood in the local base is defined, forε2>0 small, as the set V=Vc,ε2 of all c˜ that can be expressed as in (88), namely,(126)c˜σ=Φφσ,aσ=cφσ+aσNφσ,for all choices ofa:S1⟶ℝ and ϕ∈DiffS1 with(127)aCR<ε2,φ−IdCR<ε2,where(128)φ−IdCR=maxθ∈S1dS1φθ,θ+φ′θ−1+⋯+φRθ,moreover derivatives of a may be computed in arc-parameter.The above are “equivalent” in this sense. Assume that the curvec is CR+1.(i)
For anyε1 that defines neighborhood U of the first type, there is small enough ε2 that defines a neighborhood V of the second type, so that V⊆U; this is easily proved (by using Leibnitz and Faa di Bruno formulas).(ii)
Consider now a neighborhoodV of the second type, for an ε2>0 small; for ε1 small enough, the previous results Theorem 3 and Remark 12 tells us that any curve c˜∈U can be expressed in tubular coordinates; since tubular coordinates are a local diffeomorphism, similar arguments as above (plus Theorem 2) show that (for ε1 even smaller) U⊆V.We skip details for sake of brevity.In all the above there is, however, an annoying condition: to prove equivalence ofCR neighborhoods, we have to assume that the curve c is CR+1. For this reason, this works well for defining topologies in the “manifold of smooth immersed curves”; in this case, we will use neighborhoods of the first kind (or, respectively, of the second kind) for all ε>0 and all R. This is the common approach, see [12].


### 3.7.7. Local Injectivity

So far, we have consideredparametric curves. We have seen in Theorem 3 that we can represent nearby curves in a unique way using tubular coordinates, i.e., the map Φ.What happens when we considergeometric curves, that is, curves up to parameterization?Lemma 8 (local injectivity).
Letc be a C2 freely immersed planar curve. There exists a r=rc>0 such that, if(129)c˜s=defcs+asNs,c˜φs=cs+bsNs,with(130)a∞≤r,b∞≤r,∂a∂c∞≤12,∂b∂c∞≤12,where ∂/∂c is the arc derivation: then a≡b and φ is the identity.Proof.
If rescale the curvesc,c˜ and the functions a,b by a constant λ>0, and we rescale the constant rc by the same constant λ, then all hypotheses and theses are unaffected. So, we can assume with no loss of generality that c has length 2π.
If we reparameterizec^=c∘ψ then (cf the relation (93) in Remark 10) the functions a,b are reparameterized as well; having a^=a∘ψ, then ∂a^/∂c^=∂a/∂c∘ψ; and similarly for b, again hypotheses and theses are unaffected.
So, we can assume thatc is parameterized by arc parameter with no loss of generality.
Suppose that(131)c˜θ=cs+asNs,with a≤τc and a′≤1/2, then(132)c˜′=T1−κa+a′N.
So,(133)12≤c˜′≤94+14≤2.
Suppose moreover,(134)c˜φθ=cs+bsNs,with b′≤1/2, then(135)12≤c˜′φ′≤2,and then(136)14≤φ′≤4.
By contradiction, we may write(137)c˜nθ=cs+ansNs,c˜nφns=cs+bnsNs,with(138)an∞≤1n,bn∞≤1n,an′∞≤12,bn′∞≤12,where φn is not the identity: then, when 1/n<τc/4, the uniqueness condition (90) is contradicted, so there is a σn such that(139)lencn|σn,φnσn≥δc.
So, using Theorem2,(140)liminfn⟶∞dS1φn−1θn,θn>0.
Up to a subsequence, we can assume thatθn⟶θ˜ and(141)limn⟶∞dS1φn−1θn,θn=d˜>0.
We know that(142)14≤φn′≤4,up to a subsequence φn⟶φ uniformly and φn−1⟶φ−1 uniformly, where φ is a bi-Lip homeomorphism. Then, c˜n⟶c uniformly, and passing to the limit(143)c=c∘φ,so φ is a diffeomorphism. Moreover,(144)dS1φ−1θ˜,θ˜=d˜,so φ cannot be the identity; hence, c is not freely immersed.


### 3.7.8. Auto Representation (This Section May Be Skipped on a First Read, Since It Is Not Needed in the following)

The above result is very important, but the proof gives no hint on what is going on. To this end, we drop the requirement that the curve be freely immersed and look at an easy question. Is it possible for a curve to represent itself locally?Lemma 9.
Letc be a C2 immersed planar curve. There are only finitely many ways in which the curve can represent itself geometrically, that is, finitely many choices a,ϕ(145)cs+asNs=cϕs,with ϕ∈DiffS1 and a:S1⟶ℝ continuous with a∞≤τc/2. In particular, there is a ρc>0 such that, if a∞≤ρc then a≡0.In particular, ifc is freely immersed, then a∞≤ρc implies a≡0 and ϕ≡Id.Proof.
Defineδc,τc as in Definition 17 (see also Proposition 7). Let Φs,t=cs+tNs. Consider the family P of all the pairs a,φ with a∞≤τc/2 and a≢0 and(146)Φs,as=cs+asNs=cφs,we will prove that there are only finitely many such pairs.
Hence, we will letPc be smaller than the minimum of a∞ for all such pairs:(147)Pc=12mina,φ∈Pa∞.
In the example, in Figure10, there are 3 pairs in P.
We have some very strong properties.(i)
Ifa1,φ1,a2,φ2∈P and there is a s˜∈S1s.t.φ1s˜=φ2s˜, then φ1≡φ2 and a1≡a2. Indeed, there is a small interval J containing s˜, where we can invert the map Φ and(148)s,ais=Φ−1cφis,that is, the first component ofΦ−1∘c is φi−1, so φ1≡φ2 in J; so, the set s∈S1:φ1s=φ2s is both open and closed. The previous argument also proves that a1≡a2.(ii)
Figure11 on the following page can be used as a visual guide in the following proof.LetI0⊂S1 be an open interval such that the length of |cI0 is less than δc and more than δc/2. As noted in Remark 9, δc≤lenc/2. Let U0 be the image of Φ for s∈I0 and t<τc. Choose ai,φi∈P with i∈1,2, let(149)Ii=φiI0,we will prove that eitherI1∩I2=∅, or I1=I2, φ1≡φ2, and a1≡a2. Assume that s˜∈I1∩I2, let si=φi−1s˜, then s1,s2∈I0; so,(150)cφisi=cs˜,using the relation (146),(151)Φsi,aisi=csi+aisiNs=cφisi=cs˜∈U,and the fact thatΦ is a diffeomorphism for s∈I0,t<τc and cs˜∈U, we obtain that s1=s2=s^ so φ1s^=φ2s^=s˜; hence, by the previous point φ1≡φ2, a1≡a2.(iii)
The differential ofΦ (computed using arc-length derivative) is(152)T1−κtds+Ndt,so the smallest principal value is at least 1/2. We can estimate the length of c for s∈Ii to be at least δc/4. Hence, there can be only finitely many such intervals.Figure 10
Example of curve with 3 pairs inP.Figure 11
Helper scheme for proof of Lemma9.Example 9.
The constantsρc in Lemma 9 and rc in Lemma 8 though cannot be estimated apriori by using differential quantities such as maxκ. These constants may arbitrarily smaller than the quantity τc (defined in Definition 17) that provides the width of the tubular neighborhood (Proposition 7). They really depend on how the curve is drawn.
This is seen in simple examples as follows:(i)
cz=εz+z2 for z∈S1⊂ℂ, or equivalently cθ=εcosθ+cos2θ,εsinθ+sin2θ for θ∈ℝ/2π∼S1that are small C∞ perturbations of the doubly traversed circle c2 seen in Example 1; see Figure 12. This curve is freely immersed, but it is quite near to the doubly traversed circle that is not freely immersed. For ε small, the curvature κc of c is approximatively 1; so, τc∼1/2, and(153)c2θ−cθ=ε,while(154)c′∼2±ε,c2′θ−c′θ=ε.
Hence,(155)c2′θ−c′θ∼ε2c′θ,so we can use Theorem 3 to express(156)c2θ=cθ+aθNθ,using tubular coordinates when ε<1/8, with a∼ε: so rc≤∼ε; similarly, we can represent cθ+π using c, so we have ρc≤∼ε.Figure 12
Curve in the Example9.


## 3.7.1. Nearby Projection

Lemma 5 (nearby projection).
Fix aCR immersed curve c, with R≥2.(1)
Ifx∈ℝ2 and σ˜∈S1 and(73)d=defx−cσ˜<δc4,then there is ana∈ℝ with a≤d and a σ∈S1 with(74)lenc|σ,σ˜≤4d,such that(75)x=cσ+aNσ.Note also thata is uniquely identified by σ.(2)
They are unique in the family ofσ,a such that a≤τc and(76)lenc|σ,σ˜≤δc,so we can seeσ,a as functions of x, as follows.(3)
Considerx˜∈ℝ2 and σ˜∈S1 for which(77)d=defx˜−cσ˜≤τc2,let ε>0 small such that d+ε<δc/4 and let B=Bx˜,ε=x∈ℝ2:x−x˜<ε for convenience. There is a choice of function a,φ:B⟶ℝ of class CR−1 such that(78)x=Φφx,ax=cφx+axNφx,for all x∈B, and they are unique as specified above.
Note thatτc/2<δc/4<τc.Proof.
Supposec is by arc parameter (with no loss of generality as explained in Remark 8); so we write (recalling (32)) a−b instead of lenc|a,b.(i)
Choosex,σ˜ as in the statement and let Jσ˜=σ˜−δc,σ˜+δc, then consider any minimum point θ^ of(79)minθ∈Jσ˜x−cθ,note that the minimum value has to be less thand; so,(80)cσ˜−cθ^≤cσ˜−x+x−cθ^≤2d,but at the same time (sinceσ˜ and θ^ are at arc distance at most δc), by the previous Proposition 6.(81)cσ˜−cθ^≥12θ^−σ˜.So, combining the two(82)4d≥θ^−σ˜,but4d<δc so θ^ is not at extremes. Then, any θ^ providing the minimum must be internal in the interval Jσ˜: by geometrical reasoning the segment from x to cθ^ is orthogonal to the curve so there is a a such that(83)x=Φθ^,a.(ii)
Recall thatδc/4≤τc; the map Φ is injective for θ∈Jσ˜ and a≤τc, so θ^,a are unique.(iii)
For anyx∈B, we have(84)x−cσ˜<d+ε,and since d+ε<δc/4, then there is an unique σ∈Jσ˜ and a with a≤τc such that(85)x=cσ+aNσ,and we denote them by σ=φx,a=ax. Moreover, we can invert the function(86)Φ:Jσ˜×−τc,τc⟶ℝ2,and write(87)φx,ax=Φ−1x,for x∈B. This proves that φ,a∈C1B.


## 3.7.2. Global Lifting

Proposition 9 (global lifting).
Supposec:S1⟶ℝ2 is a CR immersed curve and c˜:S1⟶ℝ2 is CR−1, with R≥2. Fix 0<τ<δc/4. Suppose that we have c˜θ−cθ≤τ for all θ. There exists choice of a:S1⟶ℝ and φ:S1⟶S1 such that(88)c˜σ=Φφσ,aσ=cφσ+aσNcφσ,∀σ∈S1,with aσ≤τ and(89)lenc|σ,φσ≤4τ,holding for all σ. And, they are unique in the class of CR−1 functions such that a≤τc and(90)lenc|σ,φσ≤δc.Note also thata is uniquely identified by φ.Proof.
We just substitutex=c˜σ in the previous Lemma. By the second point, we can define functions φθandaθ uniquely as prescribed. By the third point, they are C1.Remark 10.
Suppose we are given two curvesc1,c2 and we know that there exists a choice a,φ such that(91)c2σ=c1φσ+aσNc1φσ,∀σ∈S1,as in (88). If we rotate or translate the two curves, then the above relation will hold, with the same a,φ. If we rescale the two curves by λ>0, then the relation will hold with λa,φ.
If we chooseψ∈Diff+S1 and we reparameterize all curves at the same time by c˜i=ci∘ψ, then(92)c˜2s=c˜1φ˜s+a˜sNc˜1φ˜s,∀s∈S1,holds for(93)a˜=a∘ψ,φ˜=ψ−1∘φ∘ψ.
This follows from direct computation and Remark5.Example 7.
So far so good, butφ may fail to be a diffeomorphism, as in this simple example in Figure 7. where the curve c is blue and the curve c˜ is red.But some simple lemmas can help.Figure 7
Curves in Example7; curve c is blue and curve c˜ is red.Figure 8
Scheme for Lemma 6.Lemma 6.
Let0<α<1. If we have w,v∈ℝn such that(94)w−v≤αv,then the angle β between v and w satisfies β≤arcsinα, and moreover,(95)1−αv≤v⋅w≤1+αv,1−αv≤w≤1+αv.See Figure 8 on the next page.Remark 11.
Letα>0. Suppose c1,c2,c3:S1⟶ℝ2 are C1 maps and(96)c2′θ−c3′θ≤αc1′θ,if ψ∈Diff+S1 and we reparameterize all curves at the same time by c˜i=ci∘ψ, then(97)c˜2′θ−c˜3′θ≤αc˜3′θ.
Similarly, if we rescale, rotate, or translate all curves at the same time.So, (96) is a geometric estimate, indeed we may rewrite it as(98)∂c1c2θ−∂c1c3θ≤α,where ∂c1=1/c1′∂/∂θ.Lemma 7.
Assume all hypotheses in Proposition 9. Assume moreover thatτ≤τc/4 and assume that(99)c˜′σ−c′σ≤12c′σ,∀σ∈S1,then c˜ is immersed and φ is a diffeomorphism. Moreover, when c is parameterized at constant speed, we can state that(100)15≤φ′≤4.Proof.
We rescale and reparameterizec to arc parameter using a reparameterization ψ, and at the same time, we rescale and reparameterize c˜ using the same rescaling and ψ (note that c˜ is not necessarily by arc parameter); with no loss in generality, as explained in Remarks 10 and 11.
Letβ be the angle between c˜′σ and c′σ: by Lemma 6, β=arcsinα, so it is at most π/6. Let θ=φσ; we know that σ−θ≤4τ≤τc; the angle γ between c′σ and c′θ is at most σ−θmaxk so γ≤1/2: so the angle β+γ between c˜′σ and c˜′φσ is at most 1/2+π/6, and this is less than π/2.
Deriving inσ and assuming that c is by arc parameter(101)c˜′σ=Tφ′σ1−κa+a′N,where T,N,κ are evaluated at φσ; then,(102)c˜′⋅T=φ′σ1−κa,now if a≤τc then 1−κa≥1/2; moreover, by the above reasoning c˜′⋅T>0, so φ′σ>0. Moreover, we note that 1/2≤c˜′≤3/2, 1/2≤1−κa≤3/2 and cos1/2+π/6≤6/10 to prove (100). Relation (93) tells then that φ will always be a diffeomorphism, for any curve satisfying the hypotheses.We summarize all the above: we show sufficient hypothesis such thatc˜ may be represented intubular coordinates aroundc.Theorem 3 (representation theorem).
Supposec:S1⟶ℝ2 is a CR immersed curve and c˜:S1⟶ℝ2 is CR−1, with R≥2. Define δc,τc as in Definition 17. Fix 0<τ≤τc/4. Suppose that we have c˜θ−cθ≤τ and(103)c˜′θ−c′θ≤c′θ2,for all θ.
Then,c˜ is immersed, there are φ∈Diff+S1 and a:S1⟶−τ,τ of class CR−1 such that(104)c˜φθ=Φθ,aθ=cθ+aθNθ,∀θ,with aσ≤τ and(105)lenc|σ,φσ≤4τ,holding for all σ.
They are unique in the class ofC1 functions such that a≤τc and(106)lenc|θ,φθ≤δc.Note also thata is uniquely identified by φ.Proof.
We can rescale and reparameterizec to arc parameter, and we rescale and reparameterize c˜ at the same time (c˜ will not be by arc parameter in general); as discussed in Remarks 8, 10, and 11, the hypotheses and theses are unaffected by this action. Then we apply all previous results. Just note that(107)maxθ∈S1θ−φθ=maxθ∈S1θ−φ−1θ,for any diffeomorphism.Remark 12.
Actually, rerunning on the above proofs with some patience, we can improve the above thesis a bit. We add to the previous theorem these hypotheses: fix0<α≤1/2 and then 0<τ≤ατc/2 and suppose that we have c˜θ−cθ≤τ and(108)c˜′θ−c′θ≤αc′θ2,for all θ.
Then, all above thesis hold, moreover, there are two continuous functionsf,g:0,∞⟶ℝ (independent on α,τ) with f0=g0=1, such that(109)fα≤φ′≤gα.


## 3.7.3. Asymmetry

Warning: the previous theorem seems symmetric, but it is not. The caveat is in the constantsτc,δc: it may be the case that they are quite different from τc˜,δc˜. In Figure 9, we see a piece of the two curves: curve c is blue and is flat; curve c˜ is red, and it has two inflections points A,B where the tangents are at an angle β which is as small as we would like, but then the inflection points can also be so close that the normals will cross before reaching curve c. So, while there is an easy way of representing c˜ using tubular coordinates around c, there is no way to find φ˜∈DiffS1,a˜ so as to write(110)cφθ=c˜θ+a˜θN˜θ.Figure 9
Curves for the example in Section3.7.3; curve c is blue and curve c˜ is red.


## 3.7.4. Vice Versa

We have also a sort ofvice versa of the previous Theorem 3.Proposition 10 (derepresentation).
Supposec:S1⟶ℝ2 is a C2 immersed curve and c1,c2:S1⟶ℝ2 are given by tubular coordinates(111)ciθ=Φθ,aiθ=cθ+aiθNθ,where i=1,2 and a1,a2:S1⟶ℝ are of class C1; define(112)α=defa1−a2∞maxκ,β=def∂∂ca1−∂∂ca2∞,(where κ is the curvature of c). Then,(113)c1′θ−c2′θ≤c′θα2+β2,and (obviously) c1−c2∞=2ατc, for all θ.Proof.
We rescale and reparameterizec by arc parameter, and we rescale and reparameterize c1,c2 along with c, as explained in Remark 11; this operation is justified by Remarks 8 and 10; in particular, note that α,β are scale invariant; then,(114)ci′=T1−κai+ai′N,so(115)c1′−c2′=−Tκa1−a2+a1′−a2′N.
Hence,(116)c1′−c2′≤α2+β2.


## 3.7.5. Loss of Regularity

Unfortunately, the representation discussed above suffers from a loss of regularity. Indeed, ifc:S1⟶ℝ2 is C2 and a:S1⟶ℝ is C∞, it may be the case that(117)c˜θ=Φθ,aθ=cθ+aθNθ,is of class C1 but not of class C2.This can be seen in very simple examples.Example 8.
Suppose that, fort near t=0, we have(118)ct=t,0,t<0,t,t33,t≥0,such a curve is C2 but not C3; then, for t≥0,(119)Tt=11+t41,t2,Nt=11+t4−t2,1,κ=2t1+t43/2,(that can be easily computed using a standard formula for curvature of planar curves, see Sec. 1.7.1 in [21]).
Choose thena≡1, so(120)c˜t=ct+Nt,but then(121)c˜t″=ct″+Nt″,and for t>0,(122)N″t=25t4−1t4+15/2,6t2t4−1t4+15/2.
So,(123)limt⟶0+c˜t″=−2,0,but c˜t″=0 for t<0.


## 3.7.6. Neighborhoods of a Curve

The above results encode two different but equivalent ways to define a topology on the “manifold of immersed curves.” We specify them by describing the local bases of neighborhoods of a curvec.(i)
The “Banach way” in which a local base of open neighborhoods of a curvec is given by the sets(124)U=Uc,ε1=c˜:S1⟶ℝ2:c−c˜CR<ε1,whereε1>0 is small, and(125)cCR=defmaxθ∈S1cθ+c′θ+⋯+cR.(ii)
The “geometric way” in which a neighborhood in the local base is defined, forε2>0 small, as the set V=Vc,ε2 of all c˜ that can be expressed as in (88), namely,(126)c˜σ=Φφσ,aσ=cφσ+aσNφσ,for all choices ofa:S1⟶ℝ and ϕ∈DiffS1 with(127)aCR<ε2,φ−IdCR<ε2,where(128)φ−IdCR=maxθ∈S1dS1φθ,θ+φ′θ−1+⋯+φRθ,moreover derivatives of a may be computed in arc-parameter.The above are “equivalent” in this sense. Assume that the curvec is CR+1.(i)
For anyε1 that defines neighborhood U of the first type, there is small enough ε2 that defines a neighborhood V of the second type, so that V⊆U; this is easily proved (by using Leibnitz and Faa di Bruno formulas).(ii)
Consider now a neighborhoodV of the second type, for an ε2>0 small; for ε1 small enough, the previous results Theorem 3 and Remark 12 tells us that any curve c˜∈U can be expressed in tubular coordinates; since tubular coordinates are a local diffeomorphism, similar arguments as above (plus Theorem 2) show that (for ε1 even smaller) U⊆V.We skip details for sake of brevity.In all the above there is, however, an annoying condition: to prove equivalence ofCR neighborhoods, we have to assume that the curve c is CR+1. For this reason, this works well for defining topologies in the “manifold of smooth immersed curves”; in this case, we will use neighborhoods of the first kind (or, respectively, of the second kind) for all ε>0 and all R. This is the common approach, see [12].


## 3.7.7. Local Injectivity

So far, we have consideredparametric curves. We have seen in Theorem 3 that we can represent nearby curves in a unique way using tubular coordinates, i.e., the map Φ.What happens when we considergeometric curves, that is, curves up to parameterization?Lemma 8 (local injectivity).
Letc be a C2 freely immersed planar curve. There exists a r=rc>0 such that, if(129)c˜s=defcs+asNs,c˜φs=cs+bsNs,with(130)a∞≤r,b∞≤r,∂a∂c∞≤12,∂b∂c∞≤12,where ∂/∂c is the arc derivation: then a≡b and φ is the identity.Proof.
If rescale the curvesc,c˜ and the functions a,b by a constant λ>0, and we rescale the constant rc by the same constant λ, then all hypotheses and theses are unaffected. So, we can assume with no loss of generality that c has length 2π.
If we reparameterizec^=c∘ψ then (cf the relation (93) in Remark 10) the functions a,b are reparameterized as well; having a^=a∘ψ, then ∂a^/∂c^=∂a/∂c∘ψ; and similarly for b, again hypotheses and theses are unaffected.
So, we can assume thatc is parameterized by arc parameter with no loss of generality.
Suppose that(131)c˜θ=cs+asNs,with a≤τc and a′≤1/2, then(132)c˜′=T1−κa+a′N.
So,(133)12≤c˜′≤94+14≤2.
Suppose moreover,(134)c˜φθ=cs+bsNs,with b′≤1/2, then(135)12≤c˜′φ′≤2,and then(136)14≤φ′≤4.
By contradiction, we may write(137)c˜nθ=cs+ansNs,c˜nφns=cs+bnsNs,with(138)an∞≤1n,bn∞≤1n,an′∞≤12,bn′∞≤12,where φn is not the identity: then, when 1/n<τc/4, the uniqueness condition (90) is contradicted, so there is a σn such that(139)lencn|σn,φnσn≥δc.
So, using Theorem2,(140)liminfn⟶∞dS1φn−1θn,θn>0.
Up to a subsequence, we can assume thatθn⟶θ˜ and(141)limn⟶∞dS1φn−1θn,θn=d˜>0.
We know that(142)14≤φn′≤4,up to a subsequence φn⟶φ uniformly and φn−1⟶φ−1 uniformly, where φ is a bi-Lip homeomorphism. Then, c˜n⟶c uniformly, and passing to the limit(143)c=c∘φ,so φ is a diffeomorphism. Moreover,(144)dS1φ−1θ˜,θ˜=d˜,so φ cannot be the identity; hence, c is not freely immersed.


## 3.7.8. Auto Representation (This Section May Be Skipped on a First Read, Since It Is Not Needed in the following)

The above result is very important, but the proof gives no hint on what is going on. To this end, we drop the requirement that the curve be freely immersed and look at an easy question. Is it possible for a curve to represent itself locally?Lemma 9.
Letc be a C2 immersed planar curve. There are only finitely many ways in which the curve can represent itself geometrically, that is, finitely many choices a,ϕ(145)cs+asNs=cϕs,with ϕ∈DiffS1 and a:S1⟶ℝ continuous with a∞≤τc/2. In particular, there is a ρc>0 such that, if a∞≤ρc then a≡0.In particular, ifc is freely immersed, then a∞≤ρc implies a≡0 and ϕ≡Id.Proof.
Defineδc,τc as in Definition 17 (see also Proposition 7). Let Φs,t=cs+tNs. Consider the family P of all the pairs a,φ with a∞≤τc/2 and a≢0 and(146)Φs,as=cs+asNs=cφs,we will prove that there are only finitely many such pairs.
Hence, we will letPc be smaller than the minimum of a∞ for all such pairs:(147)Pc=12mina,φ∈Pa∞.
In the example, in Figure10, there are 3 pairs in P.
We have some very strong properties.(i)
Ifa1,φ1,a2,φ2∈P and there is a s˜∈S1s.t.φ1s˜=φ2s˜, then φ1≡φ2 and a1≡a2. Indeed, there is a small interval J containing s˜, where we can invert the map Φ and(148)s,ais=Φ−1cφis,that is, the first component ofΦ−1∘c is φi−1, so φ1≡φ2 in J; so, the set s∈S1:φ1s=φ2s is both open and closed. The previous argument also proves that a1≡a2.(ii)
Figure11 on the following page can be used as a visual guide in the following proof.LetI0⊂S1 be an open interval such that the length of |cI0 is less than δc and more than δc/2. As noted in Remark 9, δc≤lenc/2. Let U0 be the image of Φ for s∈I0 and t<τc. Choose ai,φi∈P with i∈1,2, let(149)Ii=φiI0,we will prove that eitherI1∩I2=∅, or I1=I2, φ1≡φ2, and a1≡a2. Assume that s˜∈I1∩I2, let si=φi−1s˜, then s1,s2∈I0; so,(150)cφisi=cs˜,using the relation (146),(151)Φsi,aisi=csi+aisiNs=cφisi=cs˜∈U,and the fact thatΦ is a diffeomorphism for s∈I0,t<τc and cs˜∈U, we obtain that s1=s2=s^ so φ1s^=φ2s^=s˜; hence, by the previous point φ1≡φ2, a1≡a2.(iii)
The differential ofΦ (computed using arc-length derivative) is(152)T1−κtds+Ndt,so the smallest principal value is at least 1/2. We can estimate the length of c for s∈Ii to be at least δc/4. Hence, there can be only finitely many such intervals.Figure 10
Example of curve with 3 pairs inP.Figure 11
Helper scheme for proof of Lemma9.Example 9.
The constantsρc in Lemma 9 and rc in Lemma 8 though cannot be estimated apriori by using differential quantities such as maxκ. These constants may arbitrarily smaller than the quantity τc (defined in Definition 17) that provides the width of the tubular neighborhood (Proposition 7). They really depend on how the curve is drawn.
This is seen in simple examples as follows:(i)
cz=εz+z2 for z∈S1⊂ℂ, or equivalently cθ=εcosθ+cos2θ,εsinθ+sin2θ for θ∈ℝ/2π∼S1that are small C∞ perturbations of the doubly traversed circle c2 seen in Example 1; see Figure 12. This curve is freely immersed, but it is quite near to the doubly traversed circle that is not freely immersed. For ε small, the curvature κc of c is approximatively 1; so, τc∼1/2, and(153)c2θ−cθ=ε,while(154)c′∼2±ε,c2′θ−c′θ=ε.
Hence,(155)c2′θ−c′θ∼ε2c′θ,so we can use Theorem 3 to express(156)c2θ=cθ+aθNθ,using tubular coordinates when ε<1/8, with a∼ε: so rc≤∼ε; similarly, we can represent cθ+π using c, so we have ρc≤∼ε.Figure 12
Curve in the Example9.


## 3.8. Free Immersions Are Open

We have thus come to a fundamental result.Theorem 4.
Free immersions are an open subset of immersions.We can detail and prove this fact in two ways.(i)
We can see it as a corollary of Lemma8. Consider a neighborhood V defined using the tubular coordinates; precisely, define V as in the second definition in Section 3.7.6, choosing ε2<minrc,1/2,τc/4 and R=2; knowing that c˜s≡c˜φs, we could choose a≡b, but then by Lemma 8, φ would be the identity. So, each and any curve in V is freely immersed.As discussed in Section3.7.6, this proves the result in the manifold of smooth immersed curves, where the above neighborhoods define a topology.(ii)
If we instead want to prove this for the standard BanachC2 topology (first definition in Section 3.7.6), we can proceed as follows. Suppose that cn is a sequence of immersed curves that are not free; and suppose that cn⟶c in C2. We may rescale and reparametrize all the curves so that all have length 2π and cn′≡1, and still cn⟶c in C2; we skip the details (see Theorem 7 in [20]); these assumptions simplify the following arguments. Let ϕn be a sequence such that cn=cn∘ϕn and ϕn is not the identity; as above, we know that ϕn′≡1 and we may choose ϕn to be a generator of the isotropy group, so that ϕnt=t+2π/jn; we know that jn is bounded by 4 times the curvature, so up to a subsequence jn is constant, let us call it j, and then ϕn=ϕ with ϕt=t+2π/j, and passing to the limits c∘ϕ=c, so c is not freely immersed.Note that both proofs need a compactness argument; this seems unavoidable since the size of the neighborhood cannot be estimated by using differentiable quantities, as explained in Example9.


## 4. The Manifold of Free Geometric Curves

Definition 18 (classes of curves).
(i)
ImmS1,ℝn is the class of immersed curves c: curves such that c′≠0 at all points(ii)
ImmfS1,ℝn is the class of freely immersed curve, the immersed c such that, moreover, if ϕ:S1⟶S1 is a diffeomorphism and cϕt=ct for all t, then ϕ=Id(iii)
EmbS1,ℝn are the embedded curves, maps c that are diffeomorphic onto their image cS1; and the image is an embedded submanifold of ℝn of dimension 1Each class contains the one following it (this follows from the propositions seen in Section2.3).


### 4.1. Proof of Theorem 1

Definition 19.
(157)Bi,fS1,ℝ2=ImmfS1,ℝ2Diff+S1,is the quotient of ImmfS1,ℝ2 (free immersions) by the positive diffeomorphisms DiffS1 (reparameterization).We now provide the complete proof of Theorem1, namely, that this Bi,f is a manifold, for the case of smooth freely immersed planar curves; afterward, we will show in Section 4.2 how and where the proof in [5] misses some key arguments.The following proof is for immersed curvesc:S1⟶ℝ2, in a forthcoming paper [15], we will explain how it can be generalized to the case of immersions ℐ:M⟶N.


#### 4.1.1. Quotient Topology

We discuss the topological aspect of Theorem1.Letπ:Immf⟶Bi,f be the canonical projection of the positive quotient that defines Bi,f in (157).The definition of thequotient topology is as follows. A set Z is open in Bi,f when the union of its orbits(158)π−1Z=c∈Immf:c∈Z=∪c∈Zc,is open in Immf, that is, it is open in C∞.We endowImmf with the C∞ topology described earlier and Bi,f with the induced quotient topology.Now, we want to describe a specific family of open neighborhoods that will be quite useful. Fixc1 smooth freely immersed. Let τ≤minrc1,τc1/4,1/2, where rc was defined in Lemma 8.Proposition 11.
Consider the set(159)Uc1=c˜∈Immf:c˜−c1<τ,c˜′−c1′<c1′3,and this set is open in C∞S1⟶ℝ2.Proof.
This is a simple case of the arguments of Section3.7.6. Let m=minc1′, let c2∈Uc1 and(160)α=c2−c1∞,β=∂∂c1c2−∂∂c1c1∞,we know that α<τ,β<1/3; if c3 is a smooth curve and satisfies(161)c3−c2∞<τ−α,c3′−c2′∞<13−βm,then by the results in Section 3c3∈Uc1.By Theorem4, all curves in Uc1 are freely immersed.Now, let us reparameterize all the curves inUc1 and define(162)Wc1=defc˜∘φ:c˜−c1<τ,c˜′−c1′<c1′3,φ∈Diff+,since the above conditions are reparameterization invariant, then(163)Wc1=∪c2=c1∘φ,φ∈Diff+Uc2,that is an union of open sets; hence, it is open in C∞. Moreover, it contains all the orbits of all of its curves: in the language of [5], we may say that “Wc1 is saturated for the action of Diff +S1.”So, we define(164)W˜c1=πWc1,and we have(165)Wc1=π−1W˜c1.Hence,W˜c1 is open in Bi,f.


#### 4.1.2. Geometric Representation

We discuss the representation aspect of Theorem1. Consider again the set Wc1 defined in (162). For any curve c˜ in this set, by Theorem 3, we have a representation(166)c˜∘φ=c+aN,with a≤τ.Letc^=c˜∘φ for convenience; by the derepresentation result Theorem 10 (setting α=1/8,β=1/3 in that proposition and noting 1/64+1/9<1/2), we have(167)c^′θ−c1′θ<c1′θ12,so there is an unique reparameterization of c˜ that can be expressed in tubular coordinates around c; by Lemma 8, this means that φ is uniquely identified by c˜; so, we will concentrate on a.Proposition 12.
Let(168)Qc1=defa:S1⟶ℝ:∃c˜∈Uc1,∃φ∈Diff+,c˜∘φ=c+aN,be the set of all such a. This set is open.Proof.
The mapφ,a↦c+aN∘φ−1 is smooth, and Qc1 is the projection on the second component of the counterimage of Wc1 that is open.All of the above can be stated in the language of [5] as follows: the set Wc1 is an open neighborhood of c1 in C∞, it is composed only of freely immersed curves, it is saturated for the Diff +S1-action and the map(169)φ,a↦c+aN∘φ−1,splits it smoothly as(170)Wc1=∼Qc1×Diff+S1.


#### 4.1.3. Charts

Choose a curvec1; consider the map (it is not the same map Φ defined in Proposition 7)(171)Φc1:Qc1⟶Wc1,Φc1a=defc1+aN1,and we already proved in Lemma 8 that it is injective; Φc1 is also smooth as a map from C∞ to C∞.If we compose(172)Φ˜c1=π∘Φc1,then the composition(173)Φ˜c1:Qc1⟶W˜C1,is bijective: indeed, if C1∈W˜C1, then we proved in Section 3 that, picking a c˜ in the equivalence class C1,(174)c˜∈Uc1∘φ,for an unique φ, and then(175)c˜∘φ−1=c1+aN1,for a unique a.


#### 4.1.4. Atlas

To conclude, we discuss the Atlas of charts needed for Theorem1. For i=1,2, consider now two equivalence classes Ci∈Bi,f and choose a curve ci∈Ci in each; we consider the maps(176)Φ˜ci=π∘Φci.We want to check that these are charts of an Atlas for the manifold.Suppose that(177)c˜∈W˜c1∩W˜c2,then(178)c˜φ1θ=c1θ+a˜1θNc1θ,c˜φ2θ=c2θ+a˜2θNc2θ,and we need to check that(179)Φ˜c2−1∘Φ˜c1,is smooth in a neighborhood of a˜1. We can change variable in the previous one, that is, reparameterize c1,c2, so that(180)c˜θ=c1θ+a˜1θNc1θ,c˜θ=c2θ+a˜2θNc2θ,but for a1 near a˜1 we know that(181)c1θ+a1θNc1θ∈Uc2,by Lemma 10, so by the representation Theorem 3, there are φ and a2 dependent on a1 such that(182)c1θ+a1θNc1θ=c2φθ+a2φθNc2φθ.The representation theorem’s proof shows that the dependency ofa2 on a1 is smooth (it is given by the inverse of the tubular coordinates, as discussed in the nearby projection Lemma 5).This concludes the desired proof of Theorem1.


### 4.2. Comparison with [5]

We endowN with a Riemannian metric such that scalar curvatures are bounded and convexity radius is bounded uniformly away from zero; see [7].The proof in [5] is presented for generic immersions ℐ:M⟶N; we fix one such immersion.We follow notations and definitions in [5], here copied for convenience of the user (parts copied from [5] will be in italic, and enclosed in ≪...≫).≪We choose connected open setsUαα and Wαα such that Wα⊂Wα¯⊂Uα⊂M, Wαα is an open cover of M, each Wα¯ is compact, Uαα is a locally finite open cover of M, and such that ℐ|Uα is an embedding.≫≪Letp:Nℐ⟶M be the normal bundle of ℐ, defined in the following way: for x∈M, let(183)Nℐx=defTxℐTxM⊥,that is, the orthogonal complement of tangent at ℐx of the immersed manifold in TℐxN.≫≪The following diagramis a vector bundle homomorphism overℐ, which is fiberwise injective.≫≪LetexpN:TN⟶N be the exponential map on N. Now, there is a neighbourhood U of the zero section in the previous bundle Nℐ that is small enough so that(184)expN∘ℐ¯,when restricted to U|Uα is a diffeomorphism with its image. The restriction of expN∘ℐ¯ to U is called τℐ. It will serve us as a substitute for a tubular neighborhood of ℐM.≫The notationU|Uα is not described in [5], but by its usage it should be equivalent to U∩p−1Uα. Note also that, later on, the paper adds the superscript ℐ to Uα and will write it as Uαℐ.Consider planar immersed curvesc:S1⟶ℝ2: normal vectors at cθ are a one dimensional space tNc (where Nc is the normal vector to the curve c as defined in Definition 12); hence, the fibre of Nℐ is one dimensional, so a point in the bundle Nℐ can be represented by a pair θ∈S1,t∈ℝ, and the map τℐ becomes the map Φ defined in (52) in Proposition 7. Fix a τ>0, small, as we will discuss later on. We cover S1 by arcs Uα⊆S1 each shorter than ε, and Wα⊂Uα subarcs that can be chosen so that they are an open cover; for small ε we can ensure that tubular coordinates can be perused. The open set U will include normal vectors tNc with t<ε.The statement of main Theorem 1.5 in [5] starts as follows.Letℐ be a free immersion M⟶N. Then, there is an open neighborhood Wℐ in Imm M,N which is saturated for the Diff M-action and which splits smoothly as(185)Wℐ=Qℐ×DiffM.Here,Qℐ is a smooth splitting submanifold of Imm M,N, diffeomorphic to an open neighborhood of 0 in C∞Nℐ. In particular, the space ImmfM,N is open in C∞M,N.≫The proof covers also the case whenM is not compact; we will assume that M is compact so that some arguments can be simplified.The proof goes as follows.Define(186)Uℐ=defj∈ImmM,N:jWα⊆τℐUℐ|Uαℐ∀α,(the proof then goes on showing that this is an open set—we skip details).For eachj∈Ui, we define(187)ϕij:M⟶Uℐ⊆Nℐ,ϕijx=defτℐ|Uℐ|Uαℐ−1jx,ifx∈Wαℐ.Indeed, we know thatτℐ|Uℐ|Uαℐ is a diffeomorphism onto its image, and that jx∈τℐUℐ|Uαℐ when x∈Wαℐ, by definition of Uℐ.Thenϕℐ:Uℐ⟶C∞M,Nℐ is a mapping which is bijective onto the open set(188)Vℐ=defh∈C∞M,Nℐ:hWα⊆Uℐ|Uαℐ,∀α,in C∞M,Nℐ. Its inverse is given by the smooth mapping(189)τ∗ℐ:h↦τℐ∘h.The proof then goes on showing that thisφ is smooth (we skip details).We now translate the above objects into the language of Section4.1. If c=ℐ:S1⟶ℝ2 is a freely immersed curve, and c˜=j is a curve such that(190)c˜Wα⊂Uα,then c˜−c≤2ε; so, for ε small, for each σ∈S1, there are θ∈S1,t∈ℝ such that(191)c˜σ=cθ+aNcθ,and this pair θ∈S1,t∈ℝ is exactly associated to correct point in the above bundle, that is, we can write(192)x=θ,tNcθ∈Nℐ,c˜σ=τℐx.Possibly reducing the widthε of the tubular neighborhoods, we can also use Proposition 9 to ensure that the above representation is “unique.” So, Proposition 9 can be applied, and this means that, for each σ, we can write(193)c˜σ=cφσ+aσNcφσ,so, in conclusion, we can explicitly write the map h above defined as(194)hθ=φθ,aθNcφθ,where the first component in h encodes a position in the base space M=S1 of the bundle Nℐ, the second encodes a normal vector aNc to the curve c at φθ. Indeed, the inverse τ∗ is exactly the map (193).The proof continues as follows.We haveτ∗ℐh∘f=τ∗ℐh∘f for those f∈DiffM which are near enough to the identity so that h∘f∈Vi (that was defined in (188)). We consider now the open set (we added the notation Z for ease of reference)(195)Z=defh∘f:h∈Vi,f∈DiffM⊆C∞M,Ui.Obviously, we have a smooth mapping from it into(196)C∞Uℐ×DiffM,given by(197)h↦h∘p∘h−1,p∘h,where C∞Uℐ is the space of sections of Ui⟶M.≫Here comes the first mistake in the original proof. Consider Example8, where c=ℐ is the blue curve and j=c˜ is the red curve. There is a choice of Uα,Wα such that j∈Uℐ. We can express(198)c˜σ=cφσ+aσNφσ,and then define h by (194); but when we apply the rule (197) to this h, we obtain that(199)p∘h=φ,and φ is a map φ:S1⟶S1 that is not a diffeomorphism.The correct statement is that map (197) achieves a splitting of the open set Z described in (195) into(200)C∞Uℐ×C∞M⟶M.Some conditions must be added to the definition ofZ to make sure that φ is a diffeomorphism; as was done in (159) to define U, by adding the condition c˜′−c1′<c1′/3: this condition is necessary to apply the representation Theorem 4. No similar condition is present in the proof in [5].The proof afterwards proceeds as follows.So, if we let(201)Qi=defτ∗ℐC∞Uℐ∩Vℐ⊂ImmM,N.We have(202)Wℐ=defUℐ∘DiffM=∼Qℐ×DiffM=∼C∞Uℐ∩Vℐ×DiffM,since the action of Diff M on ℐ is free. Consequently, Diff M acts freely on each immersion in Wℐ, so ImmfM,N is open in C∞M,N.(thisQi is not the same as the Qc1 defined in (168), but it has the same scope; the latter one is an open neighborhood of 0 in C∞Nℐ)This is the second mistake in the proof.Even if we restrict the open set described in (195) by adding a first order condition, so that the map (197) properly splits Q, we have not guarantee that all curves in the associated neighborhood are freely immersed. This is shown in Example 9.This is why, in our proof, we added the conditionτ≤rc to the definition of U.We have shown that the proof in [5] does not prove the desired result.


## 4.1. Proof of Theorem 1

Definition 19.
(157)Bi,fS1,ℝ2=ImmfS1,ℝ2Diff+S1,is the quotient of ImmfS1,ℝ2 (free immersions) by the positive diffeomorphisms DiffS1 (reparameterization).We now provide the complete proof of Theorem1, namely, that this Bi,f is a manifold, for the case of smooth freely immersed planar curves; afterward, we will show in Section 4.2 how and where the proof in [5] misses some key arguments.The following proof is for immersed curvesc:S1⟶ℝ2, in a forthcoming paper [15], we will explain how it can be generalized to the case of immersions ℐ:M⟶N.


### 4.1.1. Quotient Topology

We discuss the topological aspect of Theorem1.Letπ:Immf⟶Bi,f be the canonical projection of the positive quotient that defines Bi,f in (157).The definition of thequotient topology is as follows. A set Z is open in Bi,f when the union of its orbits(158)π−1Z=c∈Immf:c∈Z=∪c∈Zc,is open in Immf, that is, it is open in C∞.We endowImmf with the C∞ topology described earlier and Bi,f with the induced quotient topology.Now, we want to describe a specific family of open neighborhoods that will be quite useful. Fixc1 smooth freely immersed. Let τ≤minrc1,τc1/4,1/2, where rc was defined in Lemma 8.Proposition 11.
Consider the set(159)Uc1=c˜∈Immf:c˜−c1<τ,c˜′−c1′<c1′3,and this set is open in C∞S1⟶ℝ2.Proof.
This is a simple case of the arguments of Section3.7.6. Let m=minc1′, let c2∈Uc1 and(160)α=c2−c1∞,β=∂∂c1c2−∂∂c1c1∞,we know that α<τ,β<1/3; if c3 is a smooth curve and satisfies(161)c3−c2∞<τ−α,c3′−c2′∞<13−βm,then by the results in Section 3c3∈Uc1.By Theorem4, all curves in Uc1 are freely immersed.Now, let us reparameterize all the curves inUc1 and define(162)Wc1=defc˜∘φ:c˜−c1<τ,c˜′−c1′<c1′3,φ∈Diff+,since the above conditions are reparameterization invariant, then(163)Wc1=∪c2=c1∘φ,φ∈Diff+Uc2,that is an union of open sets; hence, it is open in C∞. Moreover, it contains all the orbits of all of its curves: in the language of [5], we may say that “Wc1 is saturated for the action of Diff +S1.”So, we define(164)W˜c1=πWc1,and we have(165)Wc1=π−1W˜c1.Hence,W˜c1 is open in Bi,f.


### 4.1.2. Geometric Representation

We discuss the representation aspect of Theorem1. Consider again the set Wc1 defined in (162). For any curve c˜ in this set, by Theorem 3, we have a representation(166)c˜∘φ=c+aN,with a≤τ.Letc^=c˜∘φ for convenience; by the derepresentation result Theorem 10 (setting α=1/8,β=1/3 in that proposition and noting 1/64+1/9<1/2), we have(167)c^′θ−c1′θ<c1′θ12,so there is an unique reparameterization of c˜ that can be expressed in tubular coordinates around c; by Lemma 8, this means that φ is uniquely identified by c˜; so, we will concentrate on a.Proposition 12.
Let(168)Qc1=defa:S1⟶ℝ:∃c˜∈Uc1,∃φ∈Diff+,c˜∘φ=c+aN,be the set of all such a. This set is open.Proof.
The mapφ,a↦c+aN∘φ−1 is smooth, and Qc1 is the projection on the second component of the counterimage of Wc1 that is open.All of the above can be stated in the language of [5] as follows: the set Wc1 is an open neighborhood of c1 in C∞, it is composed only of freely immersed curves, it is saturated for the Diff +S1-action and the map(169)φ,a↦c+aN∘φ−1,splits it smoothly as(170)Wc1=∼Qc1×Diff+S1.


### 4.1.3. Charts

Choose a curvec1; consider the map (it is not the same map Φ defined in Proposition 7)(171)Φc1:Qc1⟶Wc1,Φc1a=defc1+aN1,and we already proved in Lemma 8 that it is injective; Φc1 is also smooth as a map from C∞ to C∞.If we compose(172)Φ˜c1=π∘Φc1,then the composition(173)Φ˜c1:Qc1⟶W˜C1,is bijective: indeed, if C1∈W˜C1, then we proved in Section 3 that, picking a c˜ in the equivalence class C1,(174)c˜∈Uc1∘φ,for an unique φ, and then(175)c˜∘φ−1=c1+aN1,for a unique a.


### 4.1.4. Atlas

To conclude, we discuss the Atlas of charts needed for Theorem1. For i=1,2, consider now two equivalence classes Ci∈Bi,f and choose a curve ci∈Ci in each; we consider the maps(176)Φ˜ci=π∘Φci.We want to check that these are charts of an Atlas for the manifold.Suppose that(177)c˜∈W˜c1∩W˜c2,then(178)c˜φ1θ=c1θ+a˜1θNc1θ,c˜φ2θ=c2θ+a˜2θNc2θ,and we need to check that(179)Φ˜c2−1∘Φ˜c1,is smooth in a neighborhood of a˜1. We can change variable in the previous one, that is, reparameterize c1,c2, so that(180)c˜θ=c1θ+a˜1θNc1θ,c˜θ=c2θ+a˜2θNc2θ,but for a1 near a˜1 we know that(181)c1θ+a1θNc1θ∈Uc2,by Lemma 10, so by the representation Theorem 3, there are φ and a2 dependent on a1 such that(182)c1θ+a1θNc1θ=c2φθ+a2φθNc2φθ.The representation theorem’s proof shows that the dependency ofa2 on a1 is smooth (it is given by the inverse of the tubular coordinates, as discussed in the nearby projection Lemma 5).This concludes the desired proof of Theorem1.


## 4.1.1. Quotient Topology

We discuss the topological aspect of Theorem1.Letπ:Immf⟶Bi,f be the canonical projection of the positive quotient that defines Bi,f in (157).The definition of thequotient topology is as follows. A set Z is open in Bi,f when the union of its orbits(158)π−1Z=c∈Immf:c∈Z=∪c∈Zc,is open in Immf, that is, it is open in C∞.We endowImmf with the C∞ topology described earlier and Bi,f with the induced quotient topology.Now, we want to describe a specific family of open neighborhoods that will be quite useful. Fixc1 smooth freely immersed. Let τ≤minrc1,τc1/4,1/2, where rc was defined in Lemma 8.Proposition 11.
Consider the set(159)Uc1=c˜∈Immf:c˜−c1<τ,c˜′−c1′<c1′3,and this set is open in C∞S1⟶ℝ2.Proof.
This is a simple case of the arguments of Section3.7.6. Let m=minc1′, let c2∈Uc1 and(160)α=c2−c1∞,β=∂∂c1c2−∂∂c1c1∞,we know that α<τ,β<1/3; if c3 is a smooth curve and satisfies(161)c3−c2∞<τ−α,c3′−c2′∞<13−βm,then by the results in Section 3c3∈Uc1.By Theorem4, all curves in Uc1 are freely immersed.Now, let us reparameterize all the curves inUc1 and define(162)Wc1=defc˜∘φ:c˜−c1<τ,c˜′−c1′<c1′3,φ∈Diff+,since the above conditions are reparameterization invariant, then(163)Wc1=∪c2=c1∘φ,φ∈Diff+Uc2,that is an union of open sets; hence, it is open in C∞. Moreover, it contains all the orbits of all of its curves: in the language of [5], we may say that “Wc1 is saturated for the action of Diff +S1.”So, we define(164)W˜c1=πWc1,and we have(165)Wc1=π−1W˜c1.Hence,W˜c1 is open in Bi,f.


## 4.1.2. Geometric Representation

We discuss the representation aspect of Theorem1. Consider again the set Wc1 defined in (162). For any curve c˜ in this set, by Theorem 3, we have a representation(166)c˜∘φ=c+aN,with a≤τ.Letc^=c˜∘φ for convenience; by the derepresentation result Theorem 10 (setting α=1/8,β=1/3 in that proposition and noting 1/64+1/9<1/2), we have(167)c^′θ−c1′θ<c1′θ12,so there is an unique reparameterization of c˜ that can be expressed in tubular coordinates around c; by Lemma 8, this means that φ is uniquely identified by c˜; so, we will concentrate on a.Proposition 12.
Let(168)Qc1=defa:S1⟶ℝ:∃c˜∈Uc1,∃φ∈Diff+,c˜∘φ=c+aN,be the set of all such a. This set is open.Proof.
The mapφ,a↦c+aN∘φ−1 is smooth, and Qc1 is the projection on the second component of the counterimage of Wc1 that is open.All of the above can be stated in the language of [5] as follows: the set Wc1 is an open neighborhood of c1 in C∞, it is composed only of freely immersed curves, it is saturated for the Diff +S1-action and the map(169)φ,a↦c+aN∘φ−1,splits it smoothly as(170)Wc1=∼Qc1×Diff+S1.


## 4.1.3. Charts

Choose a curvec1; consider the map (it is not the same map Φ defined in Proposition 7)(171)Φc1:Qc1⟶Wc1,Φc1a=defc1+aN1,and we already proved in Lemma 8 that it is injective; Φc1 is also smooth as a map from C∞ to C∞.If we compose(172)Φ˜c1=π∘Φc1,then the composition(173)Φ˜c1:Qc1⟶W˜C1,is bijective: indeed, if C1∈W˜C1, then we proved in Section 3 that, picking a c˜ in the equivalence class C1,(174)c˜∈Uc1∘φ,for an unique φ, and then(175)c˜∘φ−1=c1+aN1,for a unique a.


## 4.1.4. Atlas

To conclude, we discuss the Atlas of charts needed for Theorem1. For i=1,2, consider now two equivalence classes Ci∈Bi,f and choose a curve ci∈Ci in each; we consider the maps(176)Φ˜ci=π∘Φci.We want to check that these are charts of an Atlas for the manifold.Suppose that(177)c˜∈W˜c1∩W˜c2,then(178)c˜φ1θ=c1θ+a˜1θNc1θ,c˜φ2θ=c2θ+a˜2θNc2θ,and we need to check that(179)Φ˜c2−1∘Φ˜c1,is smooth in a neighborhood of a˜1. We can change variable in the previous one, that is, reparameterize c1,c2, so that(180)c˜θ=c1θ+a˜1θNc1θ,c˜θ=c2θ+a˜2θNc2θ,but for a1 near a˜1 we know that(181)c1θ+a1θNc1θ∈Uc2,by Lemma 10, so by the representation Theorem 3, there are φ and a2 dependent on a1 such that(182)c1θ+a1θNc1θ=c2φθ+a2φθNc2φθ.The representation theorem’s proof shows that the dependency ofa2 on a1 is smooth (it is given by the inverse of the tubular coordinates, as discussed in the nearby projection Lemma 5).This concludes the desired proof of Theorem1.


## 4.2. Comparison with [5]

We endowN with a Riemannian metric such that scalar curvatures are bounded and convexity radius is bounded uniformly away from zero; see [7].The proof in [5] is presented for generic immersions ℐ:M⟶N; we fix one such immersion.We follow notations and definitions in [5], here copied for convenience of the user (parts copied from [5] will be in italic, and enclosed in ≪...≫).≪We choose connected open setsUαα and Wαα such that Wα⊂Wα¯⊂Uα⊂M, Wαα is an open cover of M, each Wα¯ is compact, Uαα is a locally finite open cover of M, and such that ℐ|Uα is an embedding.≫≪Letp:Nℐ⟶M be the normal bundle of ℐ, defined in the following way: for x∈M, let(183)Nℐx=defTxℐTxM⊥,that is, the orthogonal complement of tangent at ℐx of the immersed manifold in TℐxN.≫≪The following diagramis a vector bundle homomorphism overℐ, which is fiberwise injective.≫≪LetexpN:TN⟶N be the exponential map on N. Now, there is a neighbourhood U of the zero section in the previous bundle Nℐ that is small enough so that(184)expN∘ℐ¯,when restricted to U|Uα is a diffeomorphism with its image. The restriction of expN∘ℐ¯ to U is called τℐ. It will serve us as a substitute for a tubular neighborhood of ℐM.≫The notationU|Uα is not described in [5], but by its usage it should be equivalent to U∩p−1Uα. Note also that, later on, the paper adds the superscript ℐ to Uα and will write it as Uαℐ.Consider planar immersed curvesc:S1⟶ℝ2: normal vectors at cθ are a one dimensional space tNc (where Nc is the normal vector to the curve c as defined in Definition 12); hence, the fibre of Nℐ is one dimensional, so a point in the bundle Nℐ can be represented by a pair θ∈S1,t∈ℝ, and the map τℐ becomes the map Φ defined in (52) in Proposition 7. Fix a τ>0, small, as we will discuss later on. We cover S1 by arcs Uα⊆S1 each shorter than ε, and Wα⊂Uα subarcs that can be chosen so that they are an open cover; for small ε we can ensure that tubular coordinates can be perused. The open set U will include normal vectors tNc with t<ε.The statement of main Theorem 1.5 in [5] starts as follows.Letℐ be a free immersion M⟶N. Then, there is an open neighborhood Wℐ in Imm M,N which is saturated for the Diff M-action and which splits smoothly as(185)Wℐ=Qℐ×DiffM.Here,Qℐ is a smooth splitting submanifold of Imm M,N, diffeomorphic to an open neighborhood of 0 in C∞Nℐ. In particular, the space ImmfM,N is open in C∞M,N.≫The proof covers also the case whenM is not compact; we will assume that M is compact so that some arguments can be simplified.The proof goes as follows.Define(186)Uℐ=defj∈ImmM,N:jWα⊆τℐUℐ|Uαℐ∀α,(the proof then goes on showing that this is an open set—we skip details).For eachj∈Ui, we define(187)ϕij:M⟶Uℐ⊆Nℐ,ϕijx=defτℐ|Uℐ|Uαℐ−1jx,ifx∈Wαℐ.Indeed, we know thatτℐ|Uℐ|Uαℐ is a diffeomorphism onto its image, and that jx∈τℐUℐ|Uαℐ when x∈Wαℐ, by definition of Uℐ.Thenϕℐ:Uℐ⟶C∞M,Nℐ is a mapping which is bijective onto the open set(188)Vℐ=defh∈C∞M,Nℐ:hWα⊆Uℐ|Uαℐ,∀α,in C∞M,Nℐ. Its inverse is given by the smooth mapping(189)τ∗ℐ:h↦τℐ∘h.The proof then goes on showing that thisφ is smooth (we skip details).We now translate the above objects into the language of Section4.1. If c=ℐ:S1⟶ℝ2 is a freely immersed curve, and c˜=j is a curve such that(190)c˜Wα⊂Uα,then c˜−c≤2ε; so, for ε small, for each σ∈S1, there are θ∈S1,t∈ℝ such that(191)c˜σ=cθ+aNcθ,and this pair θ∈S1,t∈ℝ is exactly associated to correct point in the above bundle, that is, we can write(192)x=θ,tNcθ∈Nℐ,c˜σ=τℐx.Possibly reducing the widthε of the tubular neighborhoods, we can also use Proposition 9 to ensure that the above representation is “unique.” So, Proposition 9 can be applied, and this means that, for each σ, we can write(193)c˜σ=cφσ+aσNcφσ,so, in conclusion, we can explicitly write the map h above defined as(194)hθ=φθ,aθNcφθ,where the first component in h encodes a position in the base space M=S1 of the bundle Nℐ, the second encodes a normal vector aNc to the curve c at φθ. Indeed, the inverse τ∗ is exactly the map (193).The proof continues as follows.We haveτ∗ℐh∘f=τ∗ℐh∘f for those f∈DiffM which are near enough to the identity so that h∘f∈Vi (that was defined in (188)). We consider now the open set (we added the notation Z for ease of reference)(195)Z=defh∘f:h∈Vi,f∈DiffM⊆C∞M,Ui.Obviously, we have a smooth mapping from it into(196)C∞Uℐ×DiffM,given by(197)h↦h∘p∘h−1,p∘h,where C∞Uℐ is the space of sections of Ui⟶M.≫Here comes the first mistake in the original proof. Consider Example8, where c=ℐ is the blue curve and j=c˜ is the red curve. There is a choice of Uα,Wα such that j∈Uℐ. We can express(198)c˜σ=cφσ+aσNφσ,and then define h by (194); but when we apply the rule (197) to this h, we obtain that(199)p∘h=φ,and φ is a map φ:S1⟶S1 that is not a diffeomorphism.The correct statement is that map (197) achieves a splitting of the open set Z described in (195) into(200)C∞Uℐ×C∞M⟶M.Some conditions must be added to the definition ofZ to make sure that φ is a diffeomorphism; as was done in (159) to define U, by adding the condition c˜′−c1′<c1′/3: this condition is necessary to apply the representation Theorem 4. No similar condition is present in the proof in [5].The proof afterwards proceeds as follows.So, if we let(201)Qi=defτ∗ℐC∞Uℐ∩Vℐ⊂ImmM,N.We have(202)Wℐ=defUℐ∘DiffM=∼Qℐ×DiffM=∼C∞Uℐ∩Vℐ×DiffM,since the action of Diff M on ℐ is free. Consequently, Diff M acts freely on each immersion in Wℐ, so ImmfM,N is open in C∞M,N.(thisQi is not the same as the Qc1 defined in (168), but it has the same scope; the latter one is an open neighborhood of 0 in C∞Nℐ)This is the second mistake in the proof.Even if we restrict the open set described in (195) by adding a first order condition, so that the map (197) properly splits Q, we have not guarantee that all curves in the associated neighborhood are freely immersed. This is shown in Example 9.This is why, in our proof, we added the conditionτ≤rc to the definition of U.We have shown that the proof in [5] does not prove the desired result.



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*Source: 6974292-2021-12-31.xml*