# Conditions to Flatten the Curvature Tensor Associated with a Symmetric Affine Connection

Authors: Esperanza Cuellar; Ana C. Silva; David A. Lizárraga
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this study, we present a problem that consists in flattening the curvature associated with an affine connection onℝ2. This problem involves distributions and symmetric affine connections defined on a differential manifold M. For rank 1 constant distributions, we characterize the existence of solutions and get sufficient conditions to solve the problem. The general statement of the problem and some illustrative examples are given.


---

## Body

## 1. Introduction

The problem referred to asflattening the curvature of an affine connection (PFCAC) is motivated by a special case of the curvature prescription problem. The PFCAC arises in control theory by observing that there are conditions under which a mechanical system can be locally transformed (equivalent) to a simpler one and by realizing that the curvature plays an important role among those conditions.Writing the equations that describe the dynamics of a mechanical system, in a specific way, sometimes facilitates solving problems that arise in control theory (controllability, stabilization, motion planning, trajectory tracking, and optimal control), as discussed by Bullo-Lewis [1]. For instance, for a subclass of mechanical control systemQ,∇,D,Y1,…,Ym,U, where Q is a smooth n-dimensional manifold (the configuration space of the system), n is the number of degrees of freedom, ∇ is an affine connection on Q, D is a regular linear velocity constraint distribution, having the property that ∇ restricts to D can be defined, Ya:a=1,…,m is a set of vector fields on Q (the control forces), taking values in D, and U⊂ℝm is the control set. In this case, the equations of motion can be written as follows:(1)q¨i=−Γjkivjvk+uaYai,i=1,…,n.These equations are written in a coordinate chartU,ϕ of Q with coordinates q1,…,qn. Here, one considers a collection X=X1,…,Xn of vector fields as a basis for TqQ for each q∈U, then v1,…,vn are the fiber coordinates for TU in the basis X. For a control u:I⟶U, I⊂ℝ that is locally integrable, for i,j∈1,…,n, ∇∂/∂qi∂/∂qj is a vector field on u. Thus, ∇∂/∂qi∂/∂qj=Γijk∂/∂qk is a linear combination of the vector fields X on Q, for n3 uniquely defined functions Γijk:U⟶ℝ, i,j,k∈1,…,n. The Γijk are called the Christoffel symbols for ∇ in the chart U,ϕ, and one defines the type 1,3 curvature tensor R whose components can be expressed as Rbcda=∂Γdba/∂qc−∂Γcba/∂qd+ΓcmaΓdbm−ΓdmaΓcbm for a,b,c,d∈1,…,n. More details are provided in references [1–3].OnTU, system (1) can be written as follows:(2)v˙=Sv+uaYaliftv,where v takes values in D, Yalift is the vertical lift of Ya in coordinates Yalift=Yai∂/∂vi, Yai are the components of Ya, and S is the geodesic spray for ∇ in coordinates q,v on TU, and it is expressed as follows:(3)Sv=vk∂∂qk−Γijkvvivj∂∂vk.We initially focus on mechanical systems of form (1) that are locally equivalent (or transformable) to a 3-dimensional canonical form with 2 inputs u1,u2:(4)x¨1=u1,x¨2=u2,x¨3=u1x1.This system is sometimes referred to asthe extended chained form (ECF). An interesting point is that equation (4) is itself a model of controlled mechanical system [4], and affine control system can be expressed as follows:(5)z˙=Sz+u1Y1liftz+u2Y2liftz,where S is the geodesic spray for ∇ on ℝ3. In coordinates x,z on Tℝ3, it is expressed as follows:(6)Sz=∑k=46zk∂∂xk.The problem of transforming a system of form (2) into (5) may be viewed as a problem of equivalence under the relation defined by regular, static state-feedback (RSSF); that is, two systems are equivalent under RSSF if one can be transformed into the other via invertible, static state feedback of the form v,u↦z,w, where z=φv and u=αv+βvw represent, respectively, a coordinate transformation of the state variable and a change of control coordinates, with φ, a local diffeomorphism; βv, an invertible matrix for all v∈Tℝ3; α, a smooth function. The new input components are ui=αiv+βijvwj, so after the transformation one gets(7)v˙=Sv+αiv+βijvwjYiliftv,and applying the change of state variables the system becomes(8)z˙=Sz+αiz+βijzwYiliftz.If we labelS^z=Sz+αizYiliftz and Y^z=βijzYiliftz, then equation (8) can be written as ECF:(9)z˙=S^z+wY^z,where the new geodesic spray is expressed as follows:(10)S^z=z˙k∂∂zk.Comparing geodesic sprays (3) and (10), we note that the Γijk for the ECF are all zero; therefore, for the ECF, the curvature tensor is identically zero. In addition, one can observe that, after the transformation, the terms αizYiliftz have been added to the geodesic spray Sx and those terms take values in the distribution D (space generated by the input vector fields with coefficients u) so new input coefficients are effectively defined.Note thatΓijk+αijk define new functions, which we denote by Γ˜ijk, this leads us to consider the existence of a new affine connection ∇˜, determined by Γ˜ijk and expressed in terms of the original ∇ and the terms αizYiliftz, which can be naturally associated with a type 1,2 tensor F on D.Curvature may be “flattened” by getting rid of the quadratic force terms related with theΓijk, but it is not an intrinsic fact because the curvature tensor may be zero even though the Christoffel symbols are not [4]. Then, by adding functions αijk, one gets Γ˜ijk=Γijk+αijk; if we require ∇˜ to be torsionless, we need to have Γ˜ijk=Γ˜jik, but since Γijk=Γjik, one gets αijk=αjik, that is, F must be a symmetric tensor with respect to its lower indices. A more “intrinsic” approach would be to determine which D-valued tensor field F yields a new connection ∇˜XY=∇XY+FX,Y with zero curvature, which we can find out in the following proposition.Proposition 1.
LetΣ be a simple mechanical system that can be expressed as the ECF via RSSF. Then there exists a tensor field F∈ΓD⊗T∗Q⊗2 such that ∇˜=∇¯+F has associated a type (1,3) curvature tensor R˜≡0.Proof.
SinceΣ is transformable to an ECF, there is a local difeomorphism φ:Q⟶ℝ3, such that, for every q∈Q there is a neighborhood U of q such that φU is an open subset of ℝ3 and φ|U:U⟶φU is a difeomorphism. So the tangent mapping Tqφ:TqQ⟶Tφqℝ3 is a linear isomorphism for all q∈Q, provided that φq=p. Then, for every X∈ΓTQ, there is a unique vector field Y∈ΓTℝ3, which is φ related to X, that is, Yp=TqφXq, and the inverse mapping satisfies Tqφ−1Yp=Xq.
SinceQ,gQ and ℝ3,gE are Riemannian manifolds, then the pullback of gE by φ, denoted by φ∗gE, defines a Riemannian metric over Q, that is, for every v,w∈TqQ:(11)φ∗gEqv,w=gEφqTqφv,Tqφw=gEφqx,y=gE∘φqTqφ−1x,Tqφ−1y=gQqv,w,this means that φ∗gE=gQ; therefore, φ is an isometry.
Assuming that∇ denotes Levi–Civita connection of gQ and Q˜ that of gE and since φ is an isometry, φ∗∇˜=∇. On the other hand, the difference between ∇ and ∇˜ defines a tensor, that is, ∇XY−∇˜XY=FX,Y, where F is a symmetric tensor field of the type 1,2 such that F∈ΓTM⊗T∗M⊗2 [2]. In particular, if F takes values in D, that is, F∈ΓD⊗T∗M⊗2, then ∇=∇˜+F have a curvature tensor R˜, which is identically zero, of the type 1,3.
Then, a problem that consists in getting rid of the curvature arises; that is, determining if there exists a type1,2 tensor FX,Y in order to get a new connection ∇˜XY=∇XY+FX,Y with zero type 1,3 tensor curvature. All the above leads us to think that PFCAC could be written in a similar way to one kind of curvature prescription. The goal of this work is to state the PFCAC and to solve it in dimension 2 by considering constant rank 1 distributions.
The statement, details, and analysis of the PFCAC will be given in the ensuing sections of this work, and the contents of which are organized as follows: in Section2, one recalls a number of basic concepts and notations to be used in the sequel. Section 3 gives a formulation of the PFCAC, along with some examples. The materials and methods for solving the PFAC are given in Section 4. In Section 5, some results are presented for dimension 2, along with some illustrative examples. Finally, concluding remarks and possible future lines of research are given in Section 6.


## 2. Preliminary Notions

This section includes some conventions and notations that will be used in the sequel. The reader may consult references [1–3, 5–9] for more information.We use the Einstein summation convention throughout, and also we consider all geometric objects are smoothC∞, unless otherwise specified.


### 2.1. Vector Fields and Distributions

Given a manifoldM of dimension n, we denote a coordinate chart U,φ near a point x∈M, with coordinate functions x=x1,…,xn on U corresponding to φ. ΓTM denotes the set of vector fields on M. If X∈ΓTM, X assigns, to each p∈M, a tangent vector Xp∈TpM, where TpM denotes the tangent space to M at p.T∗M denotes the cotangent bundle and ΓT∗M denotes the set of 1-forms. If σ∈ΓT∗M, then σ:M⟶T∗M, and a 1-form σp∈Tp∗M is assigned to each p∈M, where Tp∗M denotes the cotangent space to M at p.A rankk distribution D on M is a mapping D, which assigns to each p∈M, a k-dimensional (vector) subspace Dp of TpM. D is called smooth if, for every point p∈M, there exists a neighborhood U of p and k smooth vector fields X1,…,Xk on U, such that Dx=spanX1…,Xk for all x∈U. A vector field X on M is said to take values in D if Xp∈Dp for all p∈M. Sometimes we abuse the notation and write X∈D.


### 2.2. Riemannian Metric

Letg be a Riemannian metric on M; g is a type 0,2 symmetric, positive-definite smooth tensor field. The tensor gp induces an inner product on TpM for each p∈M. The inner product allows one to define the notions of length and angle between vectors. In coordinates, g can be expressed as follows:(12)g=gijdxi⊗dxj,where gij are the components of a matrix-valued function with the following properties:(i)
Symmetry: forp∈M and X, Y∈TpM, gpX,Y=gpY,X(ii)
Positive-definite: forp∈M and X, Y∈TpM, gpX,X>0, whenever X≠0An Euclidian metricg˜ on ℝn is expressed in coordinates as follows:(13)g˜∂∂ri,∂∂rj=δij,where ∂/∂ri,∂/∂rj are unit vectors on ℝn and matrix form g˜=Idn×n.


### 2.3. Affine Connection

A Riemannian metric defines a unique Levi-Civita affine connection, but not every affine connection is the Levi-Civita connection of a metric [2].An affine connection on a manifoldM is defined as a mapping(14)∇:ΓTM×ΓTM⟶ΓTM,X,Y↦∇XY,where ∇XY is the covariant derivative of the vector field Y in the direction of the vector field X. The mapping satisfies the following properties:(1)
∇XaY1+bY2=a∇XY1+b∇XY2,(2)
∇fX1+gX2Y=f∇X1Y+g∇X2Y,(3)
∇XfY=f∇XY+XfY, for all a,b∈ℝ and f,g∈C∞M. Here Xf is the derivative of f in the direction of X.An affine connection∇ can be uniquely represented in terms of an ordered local frame ∂/∂xi, by the specification of n3 functions Γijk for 1≤i,j,k≤n, referred to as the Christoffel symbols, by the following expression:(15)∇∂/∂xi∂∂xj=Γijk∂∂xk,i,j=1,…,n.IfX and Y∈ΓTM are given in terms of a local frame ∂/∂xi, then X=Xi∂/∂xi, Y=Yj∂/∂xj and ∇XY can be written in coordinates as follows:(16)∇XY=∂∂xiYkXi+XiYjΓijk∂∂xk.Given an affine connection∇ on a manifold M, one readily defines two tensor fields M related to ∇, namely, a type 1,2 tensor field T:ΓTM×ΓTM⟶ΓTM called torsion, with local expression given by(17)TX,Y=∇XY−∇YX−X,Y,and a type 1,3 tensor field R:ΓTM×ΓTM×ΓTM⟶ΓTM, called curvature tensor, and given by(18)RX,Y,Z=∇X∇YZ−∇Y∇XZ−∇X,YZ.An affine connection is calledtorsion-free (or symmetric), if T=0, that is, if(19)∇XY−∇YX−X,Y=0,for allX,Y∈ΓTM,where X,Y is the Lie bracket of X and Y [2, 3, 10]. In this case, the Christoffel symbols are said to be symmetric with respect to their lower indices, that is, Γijk=Γjik. In the sequel, we shall consider only symmetric affine connections and, for the sake of simplicity, write ∂i instead of ∂/∂xi to denote the partial derivative with respect to a coordinate xi when the latter is clear from the context.


## 2.1. Vector Fields and Distributions

Given a manifoldM of dimension n, we denote a coordinate chart U,φ near a point x∈M, with coordinate functions x=x1,…,xn on U corresponding to φ. ΓTM denotes the set of vector fields on M. If X∈ΓTM, X assigns, to each p∈M, a tangent vector Xp∈TpM, where TpM denotes the tangent space to M at p.T∗M denotes the cotangent bundle and ΓT∗M denotes the set of 1-forms. If σ∈ΓT∗M, then σ:M⟶T∗M, and a 1-form σp∈Tp∗M is assigned to each p∈M, where Tp∗M denotes the cotangent space to M at p.A rankk distribution D on M is a mapping D, which assigns to each p∈M, a k-dimensional (vector) subspace Dp of TpM. D is called smooth if, for every point p∈M, there exists a neighborhood U of p and k smooth vector fields X1,…,Xk on U, such that Dx=spanX1…,Xk for all x∈U. A vector field X on M is said to take values in D if Xp∈Dp for all p∈M. Sometimes we abuse the notation and write X∈D.


## 2.2. Riemannian Metric

Letg be a Riemannian metric on M; g is a type 0,2 symmetric, positive-definite smooth tensor field. The tensor gp induces an inner product on TpM for each p∈M. The inner product allows one to define the notions of length and angle between vectors. In coordinates, g can be expressed as follows:(12)g=gijdxi⊗dxj,where gij are the components of a matrix-valued function with the following properties:(i)
Symmetry: forp∈M and X, Y∈TpM, gpX,Y=gpY,X(ii)
Positive-definite: forp∈M and X, Y∈TpM, gpX,X>0, whenever X≠0An Euclidian metricg˜ on ℝn is expressed in coordinates as follows:(13)g˜∂∂ri,∂∂rj=δij,where ∂/∂ri,∂/∂rj are unit vectors on ℝn and matrix form g˜=Idn×n.


## 2.3. Affine Connection

A Riemannian metric defines a unique Levi-Civita affine connection, but not every affine connection is the Levi-Civita connection of a metric [2].An affine connection on a manifoldM is defined as a mapping(14)∇:ΓTM×ΓTM⟶ΓTM,X,Y↦∇XY,where ∇XY is the covariant derivative of the vector field Y in the direction of the vector field X. The mapping satisfies the following properties:(1)
∇XaY1+bY2=a∇XY1+b∇XY2,(2)
∇fX1+gX2Y=f∇X1Y+g∇X2Y,(3)
∇XfY=f∇XY+XfY, for all a,b∈ℝ and f,g∈C∞M. Here Xf is the derivative of f in the direction of X.An affine connection∇ can be uniquely represented in terms of an ordered local frame ∂/∂xi, by the specification of n3 functions Γijk for 1≤i,j,k≤n, referred to as the Christoffel symbols, by the following expression:(15)∇∂/∂xi∂∂xj=Γijk∂∂xk,i,j=1,…,n.IfX and Y∈ΓTM are given in terms of a local frame ∂/∂xi, then X=Xi∂/∂xi, Y=Yj∂/∂xj and ∇XY can be written in coordinates as follows:(16)∇XY=∂∂xiYkXi+XiYjΓijk∂∂xk.Given an affine connection∇ on a manifold M, one readily defines two tensor fields M related to ∇, namely, a type 1,2 tensor field T:ΓTM×ΓTM⟶ΓTM called torsion, with local expression given by(17)TX,Y=∇XY−∇YX−X,Y,and a type 1,3 tensor field R:ΓTM×ΓTM×ΓTM⟶ΓTM, called curvature tensor, and given by(18)RX,Y,Z=∇X∇YZ−∇Y∇XZ−∇X,YZ.An affine connection is calledtorsion-free (or symmetric), if T=0, that is, if(19)∇XY−∇YX−X,Y=0,for allX,Y∈ΓTM,where X,Y is the Lie bracket of X and Y [2, 3, 10]. In this case, the Christoffel symbols are said to be symmetric with respect to their lower indices, that is, Γijk=Γjik. In the sequel, we shall consider only symmetric affine connections and, for the sake of simplicity, write ∂i instead of ∂/∂xi to denote the partial derivative with respect to a coordinate xi when the latter is clear from the context.


## 3. Problem Statement

Returning to the statement of PFCAC, given a finite dimensional manifoldM, a distribution D, and a symmetric affine connection ∇ on M, the main problem to solve is as follows:(i)
Determine a type1,2 symmetric tensor field F∈ΓD⊗T∗M⊗2 that takes values on D and such that ∇˜XY=∇XY+FX,Y has a curvature tensor R˜ that is identically zero.One easily checks that∇˜ satisfies conditions (1)–(3) of the affine connection definition.Let∂/∂xi for i=1,…,n be a basis for TM, with dxj its corresponding dual base, and D a distribution such that Dp=spanXl:l=1,…,m if F takes values in D, that is, F∈ΓD⊗T∗M⊗2, then F can be expressed as a type 1,2 tensor in the coordinate system U,φ as follows:(20)Fp=aijlXlk∂∂xk⊗dxi⊗dxj.Thus, the new affine connection∇˜ can be expressed in coordinates by the following equation:(21)∇˜∂/∂xi∂∂xjx=∇∂/∂xi∂∂xjx+Fx∂∂xi,∂∂xj,where ∇∂/∂xi∂/∂xjx=Γijkx∂/∂xi and Fx∂/∂xi,∂/∂xj=arslXlk∂/∂xk⊗dxr⊗dxs∂/∂xi,∂/∂xj; since dxr∂/∂xk=δkr one has Fx∂/∂xi,∂/∂xj=arslXlkx∂/∂xkδirδjs, equation (21) is equivalent to(22)∇˜∂/∂xi∂∂xj=Γijk+aijlXlk∂∂xk.Hence,∇˜ will be determined by the Christoffel symbols Γ˜ijk=Γijk+αijlXlk for i,j,k=1,…,n, where Γijk are the Christoffel symbols defining ∇, Xlk is the k-th component of the vector field Xl that take values in D, and αijl are the functions to be found. Indeed, the type 1,3 curvature tensor components R˜bcda associated with ∇˜ are as follows:(23)R˜bcda=Rbcda+∂cαdbrXra−∂ddαcbrXra+ΓdbμαcμηXηa+αdbrXrμΓcμa+αdbrαcμηXrμXηa−ΓcbμαdμηXηa−αcbrXrμΓdμa−αcbrαdμηXrμXηa,for a,b,c,d,μ=1,…,n and r,η=1,…,m. One sees that R˜ involves only first order partial derivatives, consequently, solving the PFCAC consists in determining conditions for the integrability of first-order of PDEs; in turn, this conditions determine the existence of the field F. The curvature tensor R˜ has n4 components, of which R˜bcda=−R˜bdca and R˜bcda=0 when c=d; therefore, resulting system of PDEs has only n2n2−1/3 independent equations with m⋅nn+1/2 unknowns.For example, inℝ2 there are 16 equations, but 4 are independent with 3 unknowns α11, α12=α21yα22. In ℝ3 there are 81 equations, 24 of which are independent, with 12 unknown α11l, α12l=α21l, α13l=α31l, α22l, α23l=α32l and α33l for l=1,2. Note that the resulting system of PDEs are overdetermined (in that there are more equations than unknowns). In the next section, sufficient conditions to solve the PFCAC are described.The resulting systems of the problem are solved by algebraic manipulation of the equations involved, in analytical form. Some characteristics related toΓijk and Rbcda in the solutions are identified and based on these, PDE system integrability conditions are found.


## 4. Results and Discussions

The following result gives a sufficient condition for solving the PFCAC for an affine connection∇ on ℝn.Proposition 2.
Letp∈ℝnandU⊂ℝbe a neighborhood ofp.
IfCx=span∇∂/∂xi∂/∂xjx:i,j=1,…,n⊆Dx for each x∈U, then exists a type (1,2) tensor field F∈ΓD⊗ℝn⊗2 such that ∇˜XY=∇XY+FX,Y has a curvature tensor R˜ of type (1,3) identically zero.Proof.
SinceU is open and p∈U, there exists a neighborhood V of p such that V⊂U and there exist vector fields X1,…,Xm∈ΓTV such that, for all x∈V, Dx=spanX1x,…,Xmx. Then by Cx⊆Dx, for each Z∈Cx, there are functions aijx such that Z=aijrxXrkx∂/∂xk, for each i,j=1,…,n, can be written as a linear combination of X1,…,Xm. In coordinates, Z can be written as ∇∂i∂jx=aijlxXlkx∂/∂xk; on the other hand, we also have ∇∂/∂xi∂/∂xjx=Γijkx∂/∂xk, which implies that Γijkx∂/∂xk=aijlxXlsx∂/∂xs. If s=k, then Γijkx∂/∂xk−aijlxXlsx∂/∂xs=0. Equivalently, Γijkx−aijlxXlsx=0 for all x∈U. If Fx≜Fijr∂/∂xr⊗dxi⊗dxj, one has Fx∂/∂xs,∂/∂xt=astlXlr∂/∂xr∈Dx; therefore, Γ˜ijk=Γijk+aijlxXls=0. Therefore, since all Γ˜ijk are identically null, the curvature tensor R˜ is identically zero.
It is possible to define the vector fieldF on all of U using partitions of the unity, as shown in reference [5]. Note that Proposition 2 only provides sufficient conditions, as we can see in the following example.Example 1.
Consider the ECF described by equation (4) with the Riemannian metricgdefined onM=x∈ℝ3:x2<1, which determines the geodesic spraySonℝ6:(24)gx=10x2010x201,Sx=x4x5x6x5x6−x2x3x4x22−1x4x6x4x5−x2x5x6x22−1,S is associated with the affine connection ∇, defined by the nonzero Christoffel symbols:(25)Γ121=Γ211=Γ233=Γ323=x22x22−1,Γ231=Γ321=Γ123=Γ213=−x22x22−1,Γ132=Γ312=−12.
Note that∇ has a nonzero curvature tensor, because(26)R1311=R1232=R3212=R3133=x24x22−1,R1122=R3322=R1133=14x22−1,R2121=R2323=x22+14x22−12,R2231=R2213=x22x22−12,and their respective skew symmetries Rbcda=−Rbdca. In this case, we have a system of PDEs with 24 equations and 12 unknowns. To flatten the curvature R, we add a symmetric field tensor of type 1,2F∈ΓD⊗ℝ∗3⊗2 to the connection ∇, here D is a distribution on ℝ3. The new connection ∇˜ is determined by the Christoffel symbols Γ˜ijk=Γijk+βijlXlk, where βijk are functions to be found and Xk are components of the vector field that takes values in the distribution D.(1)
With the distribution(27)Dx=span∂∂x1|x,∂∂x2|x.One gets(28)Fx=β1x100+β2x010,withβ1x=x52x4x2−x5x3x2+x5x1−x6x22−x6/x22−1 and β2x=2x6x4x24−2x6x4−3x52x2−x52x23/2x24−1.(2)
With the distribution(29)Dx=span∂∂x1+x2∂∂x3|x,∂∂x2|x.One gets(30)Fx=β1x10x2+β2x010,β1x=x523x1+16x3x2+3x1x22+2x3x23+x56x6x24−6x618x26−x22+x24−1,β2x=2x6x4x24−2x6x4−3x52x2−x52x232x24−1.(3)
With the distribution(31)Dx=span∂∂x1|x,∂∂x2|x+x2∂∂x3|x,it cannot be solved, since the resulting system of EDP is inconsistent.
Some questions naturally arise, such as why can the PFCAC be solved for some but not for other distributions? How much of the geodesic spray can be modified by using control vector fields that take values in the control distribution, to make the effective curvature vanish? Is there a smaller-rank distributionD such that F∈ΓD and the new Γ˜ijk are zero? With a view towards answering these questions, we perform an analysis in dimension 2, locally, around a point p∈ℝ2 with a constant rank 1 distribution D and a symmetric affine connections ∇. We refer to this problem as PFCAC2, by the Frobenius integrability theorem [6], for any constant rank 1 distribution, we can choose a vector field X∈ΓD defined on a neighborhood of p such that Xp≠0 and; therefore, there is a coordinate system U,ϕ around p such that X|U=∂/∂x1, and so, Dx=span∂/∂x1|x for all x∈U. Henceforth, we assume that the affine connection ∇ and the distribution D are expressed in a system of local coordinates U,x1,x2 on ℝ2.
In general, solving the PCAC2 is equivalent to solving the following system of PDEs:(32)∂2β11−∂1β12d1=R1121+β12∂1d1−β11∂2d1+Γ1m1β12−Γ2m1β11dm+Γ21mβ1m−Γ11mβ2md1+β1mβ21−β2mβ11d1dm,∂2β11−∂1β12d2=R1122+β12∂1d2−β11∂2d2+Γ1m2β12−Γ2m2β11dm+Γ21mβ1m−Γ11mβ2md2+β1mβ21−β2mβ11d2dm,∂2β12−∂1β22d1=R2121+β22∂1d1−β12∂2d1+Γ1m1β22−Γ2m1β12dm+Γ22mβ1m−Γ12mβ2md1+β1mβ22−β2mβ12d1dm,∂2β12−∂1β22d2=R2122+β22∂1d2−β12∂2d2+Γ1m2β22−Γ2m2β12dm+Γ22mβ1m−Γ12mβ2md2+β1mβ22−β2mβ12d2dm,where dm is the m-th component of the vector field that generates the distribution D for m=1,2.
In order to find integrability conditions of the PDEs (32), we start with a simple case, that is, when the Christoffel symbols Γijk are constant for all i,j,k=1,2, we get the following result.Proposition 3.
If∇ is determined by Christoffel symbols Γijk such that in the coordinate system U,φ all the Γijk are constant, then the PFCAC2 admits a solution.Proof.
SinceΓ122 is constant for all x∈U, let us first consider the case where Γ122≠0. Then, equation (32) is reduced to the following first-order PDE system:(33)∂2β11−∂1β12=R1121+Γ122β12−Γ112β22,(34)∂2β12−∂1β22=R2121+β11β22+Γ221+Γ222−2Γ121−β12β12+Γ111−Γ122β22,(35)0=R1122+Γ112β12−Γ122β11,(36)0=R2122+Γ112β22−Γ122β12.
Under the stated assumptions, the curvature tensorR associated with ∇ has components as follows:(37)Rbcda=ΓcmaΓdbm−ΓdmaΓcbm,fora,b,c,d=1,2.
In addition, we know thatRbcda=0 when c=d and Rbcda=−Rbdca; therefore, the nonzero components of R are as follows:(38)R1121=Γ121Γ212−Γ221Γ112,R2121=Γ111Γ221+Γ121Γ222−Γ211Γ121−Γ221Γ122,R1122=Γ112Γ211+Γ122Γ212−Γ212Γ111−Γ222Γ112,R2122==Γ112Γ221−Γ212Γ121.
PluggingR1122 and R2122 into equations (35) and (36), we get, respectively, as follows:(39)0=Γ112Γ211+Γ122Γ212−Γ212Γ111−Γ222Γ112+Γ112β12−Γ122β11,0=Γ112Γ221−Γ212Γ121+Γ112β22−Γ122β12.
Ifβ12=−Γ121 and β22=−Γ221, then equation (36) is satisfied, replacing β12 and β22 in equation (35), and when Γ122≠0, one gets β11=1/Γ212Γ122Γ122−Γ111−Γ222Γ112; now, we just need to verify that equations (33) and (34) are satisfied. Substituting R1121, R2121 and given that βij are constant functions by assumption on Γijk, then equations (33) and (34) are reduced to(40)0=Γ121Γ212−Γ221Γ112+Γ122β12−Γ112β22,0=Γ111Γ221+Γ121Γ222−Γ211Γ121−Γ221Γ122+β11β22+Γ221+Γ222−2Γ121−β12β12+Γ111−Γ122β22,in fact, β11, β12;β21, and β22 satisfy these two equations and, therefore,(41)Fx=1Γ212Γ122Γ122−Γ111−Γ222Γ11210x32−Γ12110x3x4+x4x3−Γ22110x42.
Now considering the case whereΓ122|U=0, system (32) is reduced to(42)∂2β11−∂1β12=R1121−Γ112β22,(43)∂2β12−∂1β22=R2121+β11β22+Γ221+Γ222−2Γ121−β12β12+Γ111β22,(44)0=R1122+Γ112β12,(45)0=R2122+Γ112β22,and the nonzero components of the curvature tensor are as follows:(46)R1121=−Γ221Γ112,R2121=Γ111Γ221+Γ121Γ222−Γ211Γ121,R1122=Γ112Γ211−Γ222Γ112,R2122=Γ112Γ221,by replacing R1122 and R2122 in equations (44) and (45), one gets(47)0=Γ112Γ211−Γ222Γ112+Γ112β12,0=Γ112Γ221+Γ112β22,hence, β12=−Γ121+Γ222 and β22=−Γ221. Substituting R1121, R2121, β12, and β22, then equation (19) is satisfied and from equation (42), one gets(48)∂2β11=0.
Therefore,β11=fx1 for an arbitrary function f∈C1U. Thus, β11=fx1, β12=−Γ121+Γ222, and β22=−Γ221 is a solution for systems (42)–(45), thus(49)Fx=fx110x32−Γ121−Γ22210x3x4+x4x3−Γ22110x42.
In the following case, the affine connections considered are determined by their Christoffel symbolsΓij2 such that at the least one is zero and Γij1 are all zero. In this case, sufficient conditions are obtained in the following result.Proposition 4.
Let an affine connection∇ determined by Γij2, and suppose there is a pair i,j∈1,22 such that in the coordinate system U,ϕ, Γij2|U=0, and a distribution Dx=span∂/∂x1|x for all x∈U, and the following identities are satisfied:(1)
R2122|U=0(2)
IfΓ122|U≠0 and ∂2R1122/Γ122|U=0(3)
IfΓ112|U≠0 and ∂2R1122/Γ112|U=0, then the PFAC2 admits a solutionProof.
LetDx=span∂/∂x1|x and ∇ be determined by Γij2 in the coordinate U,ϕ, then system (32) is reduced as follows:(50)∂2β11−∂1β12=Γ122β12−Γ112β22,∂2β12−∂1β22=β11β22−β122+Γ222β12−Γ122β22,0=R1122+Γ112β12−Γ122β11,0=R2122+Γ112β22−Γ122β12.(i)
Ifp∈ℝ2, then exists a neighborhood V of p with V∈Vp and V⊆U such that, for the pair 1,1∈1,22, Γ112x=0 for all x∈U, then PDE system (50) is reduced to(51)∂2β11−∂1β12=Γ122β12,(52)∂2β12−∂1β22=β11β22−β122+Γ222β12−Γ122β22,(53)0=R1122−Γ122β11,(54)0=R2122−Γ122β12,by Assumption 1 and equations (53) and (54), one gets, respectively,(55)0=R1122−Γ122β11|U,0=Γ122β12|U,from these two equations, the following cases are derived:(a)
IfΓ122x|U≠0 for all x∈U, then β12=0 and β11=R1122/Γ122, and for identity (2), equation (51) is satisfied and by equation (52), one gets −∂1β22=∂1Γ122/Γ122β22, an ordinary equation, whose primitive is β22=c0exp−∫∂1Γ122/Γ122dx1, where c0 is an integration constant, and(56)Fx=β1110x32+β2210x42.(b)
IfΓ122|U=0, then R|U≡0; therefore, equations (51)–(54) admit a trivial solution, namely, βij=0 for all i,j=1,2 and Fx≡0.(ii)
Considering the pair2,2∈1,22, Γ222x=0 for all x∈U, then PDE system (50) is reduced to(57)∂2β11−∂1β12=Γ122β12−Γ112β22,(58)∂2β12−∂1β22=β11β22−β122−Γ122β22,(59)0=R1122+Γ112β12−Γ122β11,(60)0=R2122+Γ112β22−Γ122β12,by Assumption 1 and equations (57) and (58), one gets, respectively,(61)0=R1122+Γ112β12−Γ122β11|U,0=Γ112β22−Γ122β12|U,Given these two equations, we have the following cases:(a)
IfΓ122 and Γ112 are nowhere zero on U, then β12=β22=0 and β11=R1122/Γ122 satisfy equations (56)–(58) and by Assumption 2, equation (55) is satisfied and one gets(62)Fx=R1122Γ12210x32.(b)
IfΓ122|U=0 and Γ112x|U≠0 for all x∈U, then β22=0 and β12=−R1122/Γ112 satisfy equations (56)–(58) and from equation (55), one gets ∂2β11=−∂1R1122/Γ122, an ordinary equation, whose primitive is β11=−∫∂1R1122/Γ122dx2+c1, where c1 is an integration constant, then(63)Fx=β1110x32−β1210x3x4+x4x3.(c)
IfΓ112|U=0, then R|U≡0, and therefore systems (55)–(58) admit a trivial solution, namely, βij=0 for i,j=1,2 and Fx≡0.(iii)
For the pair1,2=2,1∈1,22, Γ122∣U=0 and equation (50) is reduced to(64)∂2β11−∂1β12=−Γ112β22,(65)∂2β12−∂1β22=β11β22−β122+Γ222β12,(66)0=R1122+Γ112β12,(67)0=R2122+Γ112β22,From Assumption 1 and equations (66) and (67), one gets, respectively,(68)0=R1122+Γ112β12|U,0=Γ112β22|U,and the following cases are derived:(a)
IfΓ112|U≠0, then with β22=0 and β12=−R1122/Γ112, equations (65)–(67) are satisfied and from equation (64), one gets an ordinary equation ∂2β11=∂1R1122/Γ122 with primitive β11=−∫∂1R1122/Γ122dx2+c, where c is an integration constant, so(69)Fx=β1110x32−β1210x3x4+x4x3.(b)
IfΓ112|U=0, then R|U≡0; hence, systems (64)–(67) admit a trivial solution, this is, βij=0 for i,j=1,2 and Fx≡0.
The following case is motivated by the previous one but considering this time that the Christoffel symbolsΓij2 in the coordinate system U,x1,x2 are given by smooth arbitrary functions that depend on x1 and x2 for all i,j=1,2. In this case, there are only some combinations of the Γij2, for which the PFCAC2 admits a solution, and these are mentioned in the following result:Proposition 5.
LetD be a distribution and ∇ be an affine connection on an open subset of ℝ2, and assuming that, for a coordinate chart U,ϕ, one has Γij2|U=fx1,x2, where f∈C1ℝ2 and Dx=span∂/∂x1 for all x∈U. Then, the PFCAC2 admits solutions in the following cases:C1.IfΓ222|U=constant≠0,Γ112|U=f1x1,Γ122|U=f2x1and(i)
If satisfiedR2122|U=0 and Γ122∂2R1122−R1122∂2Γ212|U=0.C2. IfΓii2|U=Γjj2|U=xi and Γij2|U=xj for i≠j and(ii)
R212|U=0 and ∂iΓ112|U=∂jΓ122|U.C3. IfΓij2|U=xk for k=1,2 and(iii)
R1122|U=R2122|U and ∂1R1122/Γ122|U−∂2R1122/Γ122|U+R1122/Γ1222|U=0.C2. IfΓii2|U=Γjj2|U=xi and Γij2|U=xj for i≠j and(ii)
R212|U=0 and ∂iΓ112|U=∂jΓ122|U.C3. IfΓij2|U=xk for k=1,2 and(iii)
R1122|U=R2122|U and ∂1R1122/Γ122|U−∂2R1122/Γ122|U+R1122/Γ1222|U=0.(ii)
R212|U=0 and ∂iΓ112|U=∂jΓ122|U.C3. IfΓij2|U=xk for k=1,2 and(iii)
R1122|U=R2122|U and ∂1R1122/Γ122|U−∂2R1122/Γ122|U+R1122/Γ1222|U=0.C3. IfΓij2|U=xk for k=1,2 and(iii)
R1122|U=R2122|U and ∂1R1122/Γ122|U−∂2R1122/Γ122|U+R1122/Γ1222|U=0.(iii)
R1122|U=R2122|U and ∂1R1122/Γ122|U−∂2R1122/Γ122|U+R1122/Γ1222|U=0.Proof.
The PDE system resulting for this case is expressed as follows:(70)∂2β11−∂1β12=Γ122β12−Γ112β22,∂2β12−∂1β22=β11β22−β122+Γ222β12−Γ122β22,0=R1122+Γ112β12−Γ122β11,0=R2122+Γ112β22−Γ122β12.C1. If Γ222|U=constant≠0, Γ112|U=f1x1 and Γ122|U=f2x1, then the associated curvature tensor has components Rbcd1≡R2122≡0 for b,c,d=1,2, R1122≡∂1f2x1−f22x1−af1x1 and system (70) is reduced to(71)∂2β11−∂1β12=f2x1β12−f1x1β22,(72)∂2β12−∂1β22=β11β22−β122+aβ12−f2x1β22,(73)0=∂1f2x1−f22x1−af1x1+f1x1β12−f2x1β11,(74)0=f1x1β22−f2x1β12.From equations (73) and (74), one gets β11=∂1f2x1−f22x1−af1x1/f2x1+f1x1/f2x12β22 and β12=f1x1/f2x1β22, then if β22=0, one gets β12=0 and β11=∂1f2x1−f22x1−af1x1/f2x1. Note that equations (71) and (72) are also satisfied. Therefore, β12=β22=0 and β11=R1122/Γ122 are solutions for systems (71)–(74), thus(75)Fx=R1122Γ12210x32.C2. If Γii2|U=Γjj2|U=xi and Γij2|U=xj for i≠j, then the curvature tensor associated with ∇ has components Rbcd1≡R2122≡0 for b,c,d=1,2 and R1122=xj2−xi2. Thus, system (70) is reduced as follows:(76)∂2β11−∂1β12=xjβ12−xiβ22,(77)∂2β12−∂1β22=β11β22−β122+xiβ12−xjβ22,(78)0=xj2−xi2+xiβ12−xjβ11,(79)0=xiβ22−xjβ12.From this, it follows thatβ11=β22=xj and β12=xi satisfy equations (78) and (79) and by ii equations (76) and (77) also are met, thus(80)Fx=xj10x32−xi10x3x4+x4x3−xj10x42.C3. If Γij2∣U=xk, then ∇ has curvature tensor associated with components Rbcd1≡0 and R1122≡R2122. Thus, system (70) is reduced as follows:(81)∂2β11−∂1β12=xkβ12−β22,(82)∂2β12−∂1β22=β11β22−β122+xkβ12−β22,(83)0=R1122+xkβ12−β11,(84)0=R1122+xkβ22−β12.
Ifxk is not zero on U for k=1,2, then by equations (83) and (84), one gets the following equation:(85)β11=2R1122xk+β22,β12=R1122xk+β22,replacing β11 and β12 in equations (81) and (82), one gets, respectively, as follows:(86)∂2β22−∂1β22=R1122+∂1R1122xk−2∂2R1122xk,∂2β22−∂1β22=R1122−R1122xk2−∂2R1122xk.
The system has a solution if the following equation is true:(87)R1122+∂1R1122xk−2∂2R1122xk=R1122−R1122xk2−∂2R1122xk,but by identity (3)), it is satisfied; therefore, systems (81)–(84) have a solution and depend on k, that is,(i)
Ifk=1, then R1122≡R2122=1 and if x1≠0, then one gets(88)∂2β22−∂1β22=1−1x12,from which it can be deduced that ifβ22=x2−1/x1, then β11=1/x1+x2 and β12=x2 are solutions for the system and(89)Fx=1x1+x210x32+x210x3x4+x4x3+x2−1x110x42.(ii)
Ifk=2, then R1122≡R2122=−1 and if x2≠0, then one gets(90)∂2β22−∂1β22=−1+2x12,from which one can deduced that if β22=x1−2/x22, then β11=x1, β12=x1+1/x2 and, therefore, the system has a solution, thus(91)Fx=x110x32+x1+1x210x3x4+x4x3+x1−2x2210x42.
Note that in Propositions4 and 5, Γij1|U=0; however, if these are nonzero on U,ϕ and if we also consider the previous cases of Γij2, the PFCAC2 has a solution. Then, also for a connection ∇ defined by Γij1 arbitrary and Γij2 as in Propositions 4 and 5, the PFCAC2 admits a solution as follows:.Proposition 6.
LetD be a distribution and ∇ be an affine connection on ℝ2. If in the coordinate system U,ϕ, ∇ is determined by Γij1|U=fijx1,x2 and Γij2|U (Propositions 4 and 5) for i,j=1,2, and also Dx=span∂/∂x1|x for all x∈U, then, the PFCAC2 admits a solution.Proof.
Based on the assumptions of the statement, system (32) is reduced to the following PDE system:(92)∂2β11−∂1β12=R1121+Γ122β12−Γ112β22,(93)∂2β12−∂1β22=R2121+β11β22+Γ221+Γ222−2Γ121−β12β12+Γ111−Γ122β22,(94)0=R1122+Γ112β12−Γ122β11,(95)0=R2122+Γ112β22−Γ122β12.
From Proposition2, β¯ij=−Γij1 is one solution for the case, where ∇ is determined by the only nonzero Christoffel symbols Γij1, with a distribution D=span∂/∂x1|x. In this case, we have Γij2 as in Propositions 4 and 5, then we know βij˜ is a solution, from which one can infer that one solution for systems (92)–(95) is βij=βij˜−Γij1, and this is verified by substituting in each equation. By assumption, we know that the components of the curvature tensor are as follows:(96)R1121=∂1Γ121−∂2Γ111+Γ121Γ212−Γ221Γ112,R2121=∂1Γ221−∂2Γ121+Γ111Γ221+Γ121Γ222−Γ211Γ121−Γ221Γ122,R1122=∂1Γ212−∂2Γ112+Γ112Γ211+Γ122Γ212−Γ212Γ111−Γ222Γ112,R2122=∂1Γ222−∂2Γ122+Γ112Γ221−Γ212Γ121,and when replacing in equations (92)–(95), one gets(97)∂2β˜11−∂1β˜12=Γ122β˜12−Γ112β˜22,(98)∂2β˜12−∂1β˜22=β˜11β˜22−β˜122+Γ222β˜12−Γ122β˜22,(99)0=∂1Γ212−∂2Γ112+Γ122Γ212−Γ222Γ112+Γ112β˜12−Γ122β˜11,(100)0=∂1Γ222−∂2Γ122+Γ112β˜22−Γ122β˜12.
Given thatΓij2 are defined as in Propositions 4 and 5 and βij˜ is one solution, thus systems (97)–(100) can be solved and one gets(101)Fx=βij10xixj.
The following examples single out some systems of PDEs that cannot be solved. This is made clear by exposing inconsistencies among the equations.Example 2.
If∇ is defined by Γ222|U=x1 and Γij2≡0 for i,j=2,2 on U, then the curvature tensor associated has only one nonzero component, namely, R2122=1. Then, system (32) is reduced to(102)∂11=∂12,∂12=∂22+β11β22−β122+x1β12,0=1,in this case, the inconsistency is immediately evident; therefore, the system has no solution.Example 3.
If∇ is determined by Γ112|U=a, with a∈ℝ, Γ222|U=x1 and all other Christoffel symbols identically zero on U, then the curvature tensor has nonzero components. If a≠0, R1122|U=−ax1 and R2122|U=1, but if a=0, we have the previous case. Then, assuming that a≠0, the resulting system is(103)∂2β11−∂1β12=−aβ22,(104)∂2β12−∂1β22=β11β22−β122+x1β12,(105)0=−ax1+aβ12,(106)0=1+aβ22,by equations (105) and (106), one gets β12=x1, β22=−1/a and when replacing in equation (103), we obtain β11=2x2. Nevertheless, equation (104) is not satisfied because we have 0=−2x2, so the system is inconsistent and therefore has no solution.
The following example shows a particular type of inconsistency because the solvability of the system depends on a constant.Example 4.
If∇ is determined by Γ112|U=a, Γ122|U=b, for a,b∈ℝ and Γ222|U=x2, then the curvature tensor has a nonzero component R1122=b2−ax2 and one gets the system:(107)∂2β11−∂1β12=bβ12−aβ22,(108)∂2β12−∂1β22=β11β22−β122+x2β12−bβ22,(109)0=b2−ax2+aβ12−bβ11,(110)0=aβ22−bβ12,from equations (109) and (110), it follows that if b≠0, then β11=b2−ax2/b+a2/b2β22 and β12=a/bβ22 but when replacing in equations (107) and (108), one gets the following equation:(111)0=ab.
Ifa=0, systems (107)–(110) have solution β11=b and β12=β22=0 (Proposition 4, i(a); otherwise the system is inconsistent. If b=0 and a≠0, by Proposition 4, the system is inconsistent as it does not satisfy the identity (3). On the other hand, with a=b=0, ∇ has associated a zero curvature tensor and therefore the system admits trivial solution βij=0.


## 5. Conclusions and Further Research

Although the PFCAC is easy to state, solving it represents an interesting challenge, as we can see in the PFCAC2. In this study, we have studied the PFCAC2 and we realized that one of the difficult parts is to characterize or study the set of solutions of the resulting PDE system. In addition, PFCAC2 could only be solved for a few cases, with constant rank 1 distributions. The main results give only sufficient conditions for existence ofF; at this point, we are unable to determine necessary conditions.As further research, it is intended to make a more geometric interpretation, consider more general distributions, look for necessary conditions, and solve the flattening problem for higher dimensions.



---

*Source: 1528246-2022-05-06.xml*