# Smooth Approximation of Lipschitz Functions on Finsler Manifolds

Authors: M. I. Garrido; J. A. Jaramillo; Y. C. Rangel
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz functionf:M→ℝ defined on a connected, second countable Finsler manifold M, for each positive continuous function ε:M→(0,∞) and each r>0, there exists a C1-smooth Lipschitz function g:M→ℝ such that |f(x)-g(x)|≤ε(x), for every x∈M, and Lip(g)≤Lip(f)+r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebra Cb1(M) of all C1 functions with bounded derivative on a complete quasi-reversible Finsler manifold M, we obtain a characterization of algebra isomorphisms T:Cb1(N)→Cb1(M) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.


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## Body

## 1. Introduction

There are many geometrically significant functions on a Riemannian manifold which are typically Lipschitz but not smooth, as it is the case, for example, of distance functions. Thus it is interesting to study the regularization and smooth approximation of Lipschitz functions on Riemannian manifolds. This has been done in the classical work of Greene and Wu [1], where in particular it is proved that every Lipschitz real function on a (connected, second countable, and finite dimensional) Riemannian manifold can be approximated, in the C0-fine topology, by smooth Lipschitz functions whose Lipschitz constants can be made arbitrarily close to the Lipschitz constant of the original function. This result has been extended in [2] to the case of infinite-dimensional Riemannian manifolds, where some interesting applications are also given. Recently, related approximation results in the setting of the so-called Banach-Finsler manifolds have been obtained in [3].Our purpose here is to study the analogous approximation problem in the context of (finite-dimensional) Finsler manifolds, where the Finsler structure is supposed to be positively (but in general not absolutely) homogeneous. The contents of the paper are as follows. In Section2 we collect some basic preliminary facts about Finsler manifolds. Section 3 is devoted to give a mean value inequality in this context. Next, in Section 4, we obtain our main result. Namely, we prove in Theorem 8 that every Lipschitz real function on a connected, second countable Finsler manifold can be approximated, in the C0-fine topology, by C1-smooth Lipschitz functions with Lipschitz constants arbitrarily close to the Lipschitz constant of the original function. This approximation result has been used in [4] in order to obtain a version of the Myers-Nakai Theorem for reversible Finsler manifolds (that is, in the case that the Finsler structure is absolutely homogeneous). In Section 5 we introduce the class of quasi-reversible Finsler manifolds, which can be described as those Finsler manifolds where distance functions are in fact Lipschitz. As a consequence of our main result, we obtain a completeness criterium for quasi-reversible Finsler manifolds, in terms of the existence of a proper C1-smooth function with uniformly bounded derivative. In this way we extend the completeness criterium for Riemannian manifolds given by Gordon in [5]. Finally, in Section 6 we consider the normed algebra Cb1(M) of all C1 functions with bounded derivative on a quasi-reversible Finsler manifold M, and we obtain a characterization of normed algebra isomorphisms T:Cb1(N)→Cb1(M) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.


## 2. Preliminaries

We start with the basic notion of Minskowski normDefinition 1.
LetV be a finite-dimensional real vector space. One says that a functional F:V→[0,∞) is a Minkowski norm on V if the following conditions are satisfied (i)
Positivity:F(v)=0 if and only if v=0.(ii)
Triangle inequality:F(u+v)≤F(u)+F(v), for every u,v∈V.(iii)
Positive homogeneity:F(λv)=λF(v), for every v∈V and every λ>0.(iv)
Regularity:F is continuous on V and C∞-smooth on V∖{0}.(v)
Strong convexity: for everyv∈V∖{0}, the quadratic form gv associated to the second derivative of the function F2 at v, that is,
(1)gv=12d2[F2](v),is positive definite onV.We note that conditions (i) and (ii) in the above definition are, in fact, consequence of conditions (iii)–(v) (see Theorem  1.2.2 of [6]). It is clear that every norm associated to an inner product is a Minkowski norm. Recall that, in general, a Minkowski norm needs not to be symmetric, and there are indeed very interesting examples of nonsymmetric Minkowski norms, such as, for example, Randers spaces (see [6]). We say F is symmetric or absolutely homogeneous if
(2)F(λv)=|λ|F(v),foreveryv∈Vandeveryλ∈ℝ.
In this case, F is a norm in the usual sense.Now the definition of Finsler manifold is as follows.Definition 2.
AFinsler manifold is a pair (M,F), where M is a finite-dimensional C∞-smooth manifold and F:TM→[0,∞) is a continuous function defined on the tangent bundle TM, satisfying  (i)
F is C∞-smooth on TM∖{0}.(ii)
For everyx∈M, F(x,·):TxM→[0,∞) is a Minkowski norm on the tangent space TxM.In particular, a Riemannian manifold is a special case of Finsler manifold, where the Minkowski norm on each tangent spaceTxM is given by an inner product. The Finsler structure F is said to be reversible if, for every x∈M, F(x,·) is symmetric. This is of course the case of Riemannian manifolds.Now suppose that(M,F) is a connected Finsler manifold. The Finsler distance  dF on M is defined by
(3)dF(x,y)=inf{ℓF(σ):σpiecewiseC1pathfromxtoy},
where the Finsler length of a piecewise C1 path σ:[a,b]→M is defined as:
(4)ℓF(σ)=∫ab‍F(σ(t),σ′(t))dt.
In this way we have (see Section 6.2 of [6]) that the Finsler distance dF is the so-called an  asymmetric distance on M, in the sense that it verifies  (i)
dF(x,y)≥0.(ii)
dF(x,y)=0 if and only if x=y.(iii)
dF(x,y)≤dF(x,z)+dF(z,y), for every x,y,z∈M.In general,dF needs not to be symmetric. Nevertheless, when F is reversible the Finsler distance dF is symmetric, and therefore (M,dF) is a metric space in the usual sense. In general, for each p∈M and r>0, the  forward ball of center p and radius r is defined as
(5)Bp+(r)={x∈M:dF(p,x)<r}.
In the same way, the  backward ball of center p and radius r is defined as
(6)Bp-(r)={x∈M:dF(x,p)<r}.
Note that, as can be seen in [6], the family of forward balls and also the family of backward balls are both neighborhood basis for the topology of the manifold M.If(V,F) is a  Minkowski space, that is, a vector space endowed with a Minkowski norm, then the associated asymmetric distance dF is given by
(7)dF(u,v)=F(v-u).
In this case, we will denote by B0(r) the forward ball of center 0∈V and radius r, and we call it the  Minkowski ball of center 0 and radius r. That is,
(8)B0(r)=B0+(r)={v∈V:F(v)<r}.We next recall the following result by Deng and Hou (see Theorem  1.2 in [7]) concerning the exponential mapping in a Finsler manifold, which will be useful in what follows.Theorem 3 (Deng and Hou [7]).
Let(M,F) be a connected Finsler manifold, let x∈M, and consider r>0 such that the exponential mapping expx:B0(r)→Bx+(r) is a C1-diffeomorphism. Then, for u,v∈B0(r) with u≠v, one has that
(9)lim(u,v)→(0,0)F(x,v-u)dF(expx(u),expx(v))=1.A terminological remark is now in order. Suppose thatX and Y are two nonempty sets, endowed with asymmetric distances dX and dY, respectively. We say that a mapping f:(X,dX)→(Y,dY) is  Lipschitz, with constant C>0 (or, briefly, C-Lipschitz) if, for every x1,x2∈X,
(10)dY(f(x1),f(x2))≤C·dX(x1,x2).
As usual, we will say that f:(X,dX)→(Y,dY) is C-bi-Lipschitz when f is bijective and both f and f-1 are C-Lipschitz mappings. With this terminology at hand and as a direct consequence of Theorem 3, we obtain the following result, describing the bi-Lipschitz behavior of the exponential mapping associated to a Finsler manifold in small balls.Corollary 4.
Let(M,F) be a connected Finsler manifold. For each x∈M and each ɛ>0, there exists r>0 such that the exponential mapping expx:B0(r)→Bx+(r) is a C1-diffeomorphism, which is (1+ɛ)-bi-Lipschitz.


## 3. Mean Value Inequality

In this section we obtain a kind of mean value inequality in the context of Finsler manifolds. If(M,F) is a connected Finsler manifold, we define the Lipschitz constant of a function f:M→ℝ as
(11)Lip(f)=sup{|f(x)-f(y)|dF(x,y):x,y∈M,x≠y}∈[0,∞].
Of course f is Lipschitz if and only if Lip(f)<∞. We denote by Lip(M) the space of all real Lipschitz functions defined on M. If f:M→ℝ is now a C1-smooth function, we define as usual the  norm of its differential df(x) at a point x∈M by
(12)∥df(x)∥F=sup{|df(x)(v)|:v∈TxM,F(x,v)≤1}.
Next, we give the following result providing the desired mean value inequality.Theorem 5.
Let(M,F) be a connected Finsler manifold and f:M→ℝ a C1-smooth function. Then,
(13)Lip(f)=sup{∥df(x)∥F:x∈M}.
Thus, for every p,q∈M, one has that
(14)|f(p)-f(q)|≤sup{∥df(x)∥F:x∈M}·dF(p,q).Proof.
For the proof, fix a numberC≥0, and we are going to see that the following conditions are equivalent:  (1)
f is C-Lipschitz.(2)
∥df(x)∥F≤C, for each x∈M.
(
1
)
⇒
(
2
) Suppose there exists some p∈M with ∥df(p)∥F>C. Then there is some v∈TpM such that F(p,v)≤1 and |df(p)(v)|>C. Suppose, for example, that df(p)(v)>C, the other case being analogous. As it is shown [6] (see Theorem  6.3.1), we can choose r>0 such that the geodesic γ(t)=expp(tv), defined for t∈[0,r], minimizes the Finsler distance from point p, that is, dF(p,γ(s))=s, for every s∈[0,r]. Now define h:[0,r]→ℝ by h(t)=f(γ(t)). We then have that h′(0)=df(p)(v)>C, and therefore there exists some δ∈(0,r) such that
(15)h(t)-h(0)t>C,if0<t≤δ.
Since p=γ(0), choosing q=γ(δ), we obtain that
(16)|f(p)-f(q)|=h(δ)-h(0)>C·δ=C·dF(p,q)
which contradicts the fact that f is C-Lipschitz.
(
2
)
⇒
(
1
) Consider x,y∈M, and let ɛ>0. Choose a piecewise C1 path γ:[a,b]→M such that γ(a)=x, γ(b)=y, and
(17)ℓF(γ)≤dF(x,y)+ɛ.
Define now h:[a,b]→ℝ by h(t)=f(γ(t)). Then,
(18)|f(y)-f(x)|=|∫ab‍h′(t)dt|≤∫ab‍|h′(t)|dt=∫ab‍|df(γ(t))(γ′(t))|dt≤∫ab‍∥df(γ(t))∥F·F(γ(t),γ′(t))dt≤C∫ab‍F(γ(t),γ′(t))dt=C·ℓF(γ)≤C(dF(x,y)+ɛ).
This shows that |f(x)-f(y)|≤C·dF(x,y), for every x,y∈M.We finish this section with the following simple result giving a local characterization of Lipschitz mappings, which will be useful later.Proposition 6.
Let(M,FM) and (N,FN) be connected Finsler manifolds, with Finsler distances dM and dN, respectively. A mapping f:(M,dM)→(N,dN) is C-Lipschitz if and only if it is locally C-Lipschitz; that is, every point p∈M has a neighborhood Up such that, for every x,y∈Up,
(19)dN(f(x),f(y))≤C·dM(x,y).Proof.
Suppose thatf is locally C-Lipschitz. Consider x,y∈M and ɛ>0. Choose a piecewise C1 path γ:[a,b]→M from x to y, such that ℓM(γ)≤dM(x,y)+ɛ. Each p∈γ([a,b]) has an open neighborhood Up where f is C-Lipschitz. Choose a partition a=t0<t1<⋯<tk=b of [a,b] such that, for every i=1,…,k,  γ([ti-1,ti]) is contained into Up for some p∈γ([a,b]). Then
(20)dN(f(x),f(y))≤∑i=1k‍dN(f(γ(ti-1)),f(γ(ti)))≤C·∑i=1k‍dM(γ(ti-1),γ(ti))≤C·∑i=1k‍ℓM(γ|[ti-1,ti])=C·ℓM(γ)≤C·(dM(x,y)+ɛ).
In this way we obtain that f is C-Lipschitz. The converse is clear.


## 4. Smooth Approximation of Lipschitz Functions

In this section we present our results about regularization of Lipschitz functions on Finsler manifolds. In particular, as consequence of Theorem8 as follows, we can derive that if f:M→ℝ is a Lipschitz function defined on a connected and second-countable Finsler manifold M and ɛ>0 is given, there exists a C1-smooth Lipschitz function g:M→ℝ such that |f(x)-g(x)|≤ɛ, for every x∈M, and Lip(g)≤Lip(f)+ɛ. We start with the following simple Lemma, which gives a first result of smooth approximation in Minkowski spaces.Lemma 7.
Let(V,F) be a vector space endowed with a Minkowski norm. Consider an open set A⊂V and, for δ>0, denote
(21)Aδ={v+w:v∈A,F(w)<δ}=⋃‍v∈ABv(δ).
Suppose that f:Aδ→ℝ is Lipschitz and let ɛ>0. Then there exists a C∞-smooth function g:V→ℝ such that |f(v)-g(v)|≤ɛ, for every v∈A, and Lip(g|A)≤Lip(f|A).Proof.
By choosing a basis ofV, we may assume that V=ℝn. Note that, by local compactness, it follows that the Minkowski norm F is equivalent to the usual Euclidean norm ∥·∥ in ℝn, in the sense that there exists some R≥1 such that
(22)1R·∥v∥≤F(v)≤R·∥v∥
for every v∈ℝn. Now, if f:Aδ→ℝ is a C-Lipschitz function for the Minkowski norm F, then f is a (C·R)-Lipschitz function for the Euclidean norm. Hence, using, for example, the well-known MacShane extension result, we can obtain a Lipschitz extension f~:ℝn→ℝ.
Now consider a sequence(φk)k of usual C∞-smooth mollifiers on ℝn, where each φk is nonnegative, supp(φk) is contained in the Euclidean ball 𝔹(0,1/k), and ∫ℝn‍φk=1. For each k, define fk:ℝn→ℝ by
(23)fk(v)=∫ℝn‍f~(v+w)φk(w)dw.
Each fk is C∞-smooth, and, since f~ is uniformly continuous, we have that the sequence (fk) converges to f~ uniformly on ℝn. Given ɛ>0, choose k>R/δ and large enough so that ∥fk-f~∥∞<ɛ and define g=fk. Then, if u,v∈A,
(24)|g(u)-g(v)|=|∫ℝn‍f~(u+w)φk(w)dw-∫ℝn‍f~(v+w)φk(w)dw|≤∫ℝn‍|f~(u+w)-f~(v+w)|·|φk(w)|dw=∫𝔹(0,1/k)‍|f(u+w)-f(v+w)|·|φk(w)|dw≤C·F(v-u)·∫ℝn‍φk(w)dw=C·dF(u,v).
Therefore, we have that Lip(g|A)≤Lip(f|A), as we wanted.We next give the main result of the paper.Theorem 8.
Let(M,F) be a connected and second countable Finsler manifold, let f:M→ℝ be a Lipschitz function, consider ɛ:M→(0,+∞) a continuous function, and let r>0. Then there is a C1-smooth Lipschitz function g:M→ℝ such that |f(x)-g(x)|≤ɛ(x), for every x∈M, and Lip(g)≤Lip(f)+r.Proof.
Let us denoteC=Lip(f). Without loss of generality we may assume that, for every x∈M, ɛ(x)>0 is small enough so that ɛ(x)≤r/2 and
(25)C·(1+ɛ(x))2<C+r2.
Using Corollary 4, for each x∈M, we can choose δx>0 such that the exponential mapping expx is a C1-diffeomorphism and (1+ɛ(x))-bi-Lipschitz from the Minkowski ball B0x(3δx)⊂TxM onto the forward ball Bx+(3δx)⊂M, where 0x denotes the null vector of TxM. In addition, by the continuity of f and ɛ, we can also assume that ɛ(y)≥ɛ(x)/2 and |f(y)-f(x)|≤ɛ(x)/2, for every y∈Bx+(3δx). Since M is second countable, there is a sequence (xn) in M such that
(26)M=⋃n=1∞Bxn+(δn),
where we denote δn=δxn. Now, for each n∈ℕ, we define fn:B0xn(3δn)→ℝ by
(27)fn(v)=f(expxn(v)),
and we then have that fn is C·(1+ɛ(xn))-Lipschitz.
Next, we are going to construct a partition of unity subordinated to the covering{Bxn+(2δn)}n∈ℕ of M, estimating the Lipschitz constant of the respective functions. Thus, for each n∈ℕ, let θn:ℝ→[0,1] be a C∞-smooth function such that
(28)θn(x)={1ifx∈(-∞,δn],0ifx∈[2δn,+∞),
and define φn:M→[0,1] by
(29)φn(x)={θn(F(xn,expxn-1(x)))ifx∈Bxn+(3δn),0otherwise.
It is clear that each φn is C1-smooth and Lipschitz. Furthermore, φn=1 on the forward ball Bxn+(δn) and φn=0 on M∖Bxn+(2δn). Now we define the functions ψn:M→[0,1] by setting ψ1=φ1 and, for n≥2,
(30)ψn=φn·∏j<n‍(1-φj).
Then, it is easy to check that, for every n∈ℕ,  (i)
ψn is C1-smooth and Cn-Lipschitz, where Cn=∑j≤n‍Lip(φj).(ii)
supp(ψn)⊂supp(φn)⊂Bxn+(2δn).(iii)
ψm=0 on Bxn+(δn), whenever m>n.
Thus,{ψn} is the desired partition of unity. Indeed, for each x∈M, let n=n(x) be the first integer such that x∈Bxn+(δn). Then φn(x)=1 and ψm(Bxn+(δn))=0, for m>n. Therefore, the family {supp(ψm)}m is locally finite. In addition ∑m‍ψm(x)=1, since
(31)∑m‍ψm(x)=ψ1(x)+ψ2(x)+⋯+ψn(x)=ψ1(x)+ψ2(x)+⋯+ψn-1(x)+φn(x)∏j<n‍(1-φj)(x)=ψ1(x)+ψ2(x)+⋯+φn-1(x)×∏j<n-1‍(1-φj)(x)+∏j<n‍(1-φj)(x)=ψ1(x)+ψ2(x)+⋯+φn-2(x)×∏j<n-2‍(1-φj)(x)+∏j<n-1‍(1-φj)(x)=⋯=φ1(x)+(1-φ1(x))=1.
Now using Lemma 7 we can find, for each n, a C∞-smooth function gn:TxnM→ℝ such that
(32)|gn(v)-fn(v)|≤ɛ(xn)2n+2(Cn+1)
for every v∈B0xn(2δn), and
(33)Lip(gn|B0xn(2δn))≤Lip(fn|B0xn(2δn))≤C·(1+ɛ(xn)).
Thus, we define the approximation function g:M→ℝ by
(34)g(x)=∑n‍ψn(x)·gn(expxn-1(x))
for each x∈M. Note that, since the exponential mapping expxn is a C1-diffeomorphism from B0xn(3δn) onto Bxn+(3δn), the expression ψn(x)·gn(expxn-1(x)) is well defined for x∈Bxn+(3δn) and it is C1-smooth on Bxn+(3δn). On the other hand, if x∉Bxn+(2δn)⊃supp(ψn), then ψn(x)=0. Thus, for every x∉Bxn+(3δn), we may assume that ψn(x)·gn(expxn-1(x)) is zero. With this convention, and taking into account that {ψn} is a C1-smooth partition of unity, we obtain that g is well defined and C1-smooth on M.
We are going to see thatg is also Lipschitz and that g and Lip(g) approximate to f and Lip(f), respectively. Fix x∈M, and consider again n=n(x) the first integer such that x∈Bxn+(δn). To simplify, denote vm=expxm-1(x)∈TxmM, for all m. Then we have
(35)|g(x)-f(x)|=|∑m≤n‍ψm(x)·gm(expxm-1(x))-f(x)|=|∑m≤n‍ψm(x)·(gm(vm)-f(x))|=|∑m≤n‍ψm(x)·(gm(vm)-fm(vm))|≤∑m≤n‍ψm(x)ɛ(xm)2m+2(Cm+1)≤∑m≤n‍ψm(x)ɛ(xm)2≤∑m≤n‍ψm(x)ɛ(x)≤ɛ(x).
Finally, let us check that g is (C+r)-Lipschitz, and hence Lip(g)≤C+r. By Proposition 6, it will suffice to see that g is locally (C+r)-Lipschitz. Fix z∈M and, as before, consider n=n(z) the first integer such that z∈Bxn+(δn). We are going to see that g is (C+r)-Lipschitz on the open set Uz=Bz+(δz)∩Bz-(δz), where
(36)δz=12min{δ1,…,δn,δn-dF(xn,z)}.
For each x,y∈M, denote
(37)Px,y={m∈{1,…,n}:Bxm+(2δm)∩{x,y}≠∅}.
Then, whenever x,y∈Uz, the following holds:(1)
ifm∈{1,…,n} and y∈Bxm+(2δm), then x∈Bxm+(3δm). Indeed, if x,y∈Uz, then dF(y,x)<2δz≤δm. Thus if y∈Bxm+(2δm) we have that
(38)dF(xm,x)≤dF(xm,y)+dF(y,x)<2δm+δm=3δm,and thereforex∈Bxm+(3δm).(2)
For everym∈Px,y we have that x,y∈Bxm+(3δm). That is clear from the above. In particular, if m∈Px,y then vm=expxm-1(x) and wm=expxm-1(y) are well defined, and, using (33) and the fact that expxm-1:Bxm+(3δm)→B0xm(3δm) is (1+ɛ(xm))-Lipschitz, we see that
(39)|gm(vm)-gm(wm)|≤C·(1+ɛ(xm))×dF(expxm-1(x),expxm-1(y))≤C·(1+ɛ(xm))2dF(x,y).(3)
Ifm∈ℕ∖Px,y then ψm(x)=0=ψm(y). That follows, since supp(ψm)⊂Bxm+(2δm) and supp(ψℓ)∩Bxn+(δn)=∅, for every l>n.
As a consequence of the above we have, for everyx,y∈Uz, and using the notation vm=expxm-1(x) and wm=expxm-1(y), the following:  (i)
g(x)=∑m∈Px,y‍ψm(x)·gm(vm),(ii)
g(y)=∑m∈Px,y‍ψm(y)·gm(wm),(iii)
∑m∈Px,y‍ψm(x)=1=∑m∈Px,y‍ψm(y),(iv)
|gm(vm)-gm(wm)|≤C(1+ɛ(xm))2dF(x,y), whenever m∈Px,y.
Therefore, since(40)∑m∈Px,y‍(ψm(x)-ψm(y))·f(x)=0,
we deduce that
(41)g(x)-g(y)=∑m∈Px,y‍ψm(x)·gm(vm)-∑m∈Px,y‍ψm(y)·gm(wm)=∑m∈Px,y‍(ψm(x)-ψm(y))·(gm(vm)-f(x))+∑m∈Px,y‍ψm(y)·(gm(vm)-gm(wm)).
Finally, using (25), (32), and the fact that ψm is Cm-Lipschitz, we then have that
(42)|g(x)-g(y)|≤∑m∈Px,y‍|ψm(x)-ψm(y)|·|gm(vm)-f(x)|+∑m∈Px,y‍ψm(y)·|gm(vm)-gm(wm)|≤∑m≤n‍Cm·ɛ(xm)(Cm+1)2m+2dF(x,y)+∑m≤n‍ψm(y)(C+r2)dF(x,y)≤∑m≤n‍ɛ(z)2m+1dF(x,y)+(C+r2)dF(x,y)≤(C+r)·dF(x,y)
since ∑m≤n‍(ɛ(z)/2m+1)≤ɛ(z)≤r/2. This shows that g is locally (C+r)-Lipschitz and we finish the proof.Remark 9.
In general, the exponential mapping in a Finsler manifold is onlyC1-smooth. According to a result of Akbar-Zadeh in [8] (see also [6], page 127), the exponential mapping is C2-smooth if and only if it is C∞-smooth, and this property characterizes a special class of Finsler manifolds, called manifolds of Berwald type. Thus, if (M,F) is a connected, second countable manifold of Berwald type, the same proof above gives that the approximating function g in Theorem 8 can be chosen to be C∞-smooth.


## 5. Quasi-Reversible Manifolds and a Completeness Criterium

In this section, as an application of the approximation result given in the above section, we obtain a completeness criterium for the class of manifolds that we call  quasi-reversible. These are defined as follows:Definition 10.
A Finsler manifold(M,F) is said to be quasi-reversible if there exists some C≥1 such that
(43)F(x,-v)≤C·F(x,v),forevery(x,v)∈TM.It is clear that every reversible Finsler manifold is quasi-reversible. In fact, a Finsler manifold is reversible if and only if it is quasi-reversible forC=1. On the other hand, a remarkable class of quasi-reversible (not necessarily reversible) manifolds are those manifolds of Berwald type. Indeed, we can deduce this, using a result due to Ichijyō [9] (see also [6], page 258) saying that if M is a manifold of Berwald type, then all its tangent spaces (TxM,F(x,·)), for every x∈M, are linearly isometric to each other.We next give a useful characterization of connected quasi-reversible manifolds.Theorem 11.
Let(M,F) be a connected Finsler manifold, and let C≥1. The following conditions are equivalent:  (1)
F(x,-v)≤C·F(x,v), for every (x,v)∈TM.(2)
(1/C)·dF(y,x)≤dF(x,y)≤C·dF(y,x), for every x,y∈M.(3)
For allp∈M, the forward distance function Φp=dF(p,·) is C-Lipschitz.(4)
For allp∈M, the backward distance function Ψp=dF(·,p) is C-Lipschitz.Proof.
(
1
)
⇒
(
2
) Let x,y∈M and consider γ:[0,1]→M a piecewise C1 path from y to x. Then the reverse path γ~:[0,1]→M given by γ~(t)=γ(1-t) is a piecewise C1 path from x to y. Now,
(44)ℓF(γ~)=∫01‍F(γ~(t),γ~′(t))dt=∫01‍F(γ(1-t),-γ′(1-t))dt=∫01‍F(γ(s),-γ′(s))ds≤C·∫01‍F(γ(s),γ′(s))ds=C·ℓF(γ).
This implies that dF(x,y)≤C·dF(y,x). Interchanging the roles of x and y, we obtain the reverse inequality.
(
2
)
⇒
(
1
) Let x∈M. From Theorem 3 we have that if v≠0 then
(45)limt→0+F(x,tv-t(-v))dF(expx(t(-v)),expx(tv))=1=limt→0+F(x,t(-v)-tv)dF(expx(tv),expx(t(-v))).
That is,
(46)limt→0+F(x,2tv)dF(expx(-tv),expx(tv))=1=limt→0+F(x,2t(-v))dF(expx(tv),expx(-tv)).
Thus we obtain that
(47)F(x,-v)=limt→0+dF(expx(tv),expx(-tv))2t≤Climt→0+dF(expx(-tv),expx(tv))2t=C·F(x,v).(2)⇒(3) For every p,x,y∈M we have, from the triangle inequality, that
(48)dF(p,x)-dF(p,y)≤dF(y,x)dF(p,y)-dF(p,x)≤dF(x,y).
By the hypothesis (2), we follow at once that
(49)|dF(p,x)-dF(p,y)|≤C·dF(x,y).
This means that the function Φp=dF(p,·) is C-Lipschitz.
(
3
)
⇒
(
2
) For every p,x,y∈M we have that, in particular,
(50)dF(p,x)-dF(p,y)≤C·dF(x,y).
Choosing p=y we have that
(51)dF(y,x)≤C·dF(x,y).
Reversing the roles of x and y, we also have that dF(x,y)≥(1/C)·dF(y,x).
(
4
)
⇔
(
2
) This can be seen as before.Note that, by choosingC=1 in the above result, we can deduce at once the following characterization of connected reversible manifolds.Corollary 12.
Let(M,F) be a connected Finsler manifold. The following conditions are equivalent:  (1)
F(x,-v)=F(x,v), for every (x,v)∈TM,(2)
dF(y,x)=dF(x,y), for every x,y∈M.As an application of Theorem8, we are going to obtain a completeness criterium in the context of quasi-reversible manifolds. This will extend the corresponding result by Gordon [5] for Riemannian manifolds. First recall that a sequence (xn) in a Finsler manifold (M,F) is said to be  forward Cauchy (respectively,  backward Cauchy) if, for every ɛ>0, there exists some n0∈ℕ such that, if n0≤n≤m, then dF(xn,xm)<ɛ, (respectively, dF(xm,xn)<ɛ). We say then that (M,F) is  forward complete (respectively,  backward complete) if every forward Cauchy sequence is convergent (respectively, every backward Cauchy sequence is convergent). It is clear that, for quasi-reversible manifolds, forward and backward completeness are equivalent. On the other hand, recall that a continuous function f:M→ℝ is said to be  proper if, for every compact set K⊂ℝ, its preimage f-1(K) is compact.Theorem 13.
Let(M,F) be a connected, second countable, and quasi-reversible Finsler manifold. The following conditions are equivalent:  (1)
(M,F) is forward complete.(2)
There exists a proper Lipschitz functionf:M→ℝ.(3)
There exists a properC1-smooth function g:M→ℝ whose differential is uniformly bounded in norm.Proof.
(
1
)
⇒
(
2
) Fix p∈M and consider the forward distance function φp=dF(p,·). The Hopf-Rinow Theorem (see Theorem  6.6.1 in [6]) gives that (M,F) is forward complete if and only if every closed and (forward) bounded subset of M is compact. This implies that φp is a proper function. Furthermore, by Theorem 11, we have that φp is Lipschitz.
(
2
)
⇒
(
3
) Suppose that the proper function f:M→ℝ is C-Lipschitz, and fix some ɛ>0. By Theorem 8, there exists a C1-smooth function g:M→ℝ such that  (i)
|f(x)-g(x)|≤ɛ, for every x∈M.(ii)
Lip(g)≤C+ɛ.
It is easy to check thatg is a proper function, since f is so. On the other hand, by Theorem 5, we have that ∥dg(x)∥F≤C+ɛ, for every x∈M.
(
3
)
⇒
(
1
) Let (xn) be a forward Cauchy sequence in M. Since g:M→ℝ is Lipschitz, then {g(xn)} is a Cauchy sequence in ℝ, and therefore {g(xn)} converges to some point z in ℝ. Thus K={z}∪{g(xn):n∈ℕ} is a compact subset of ℝ. Now (xn) is contained in g-1(K), which is compact since g is proper. Then (xn) is convergent in M.


## 6. Algebras of Differentiable Functions on Finsler Manifolds

The classical Myers-Nakai Theorem asserts that the Riemannian structure of a Riemannian manifoldM is determined by the natural normed algebra structure on the space Cb1(M) of all bounded C1 functions on M which have bounded derivative (or, equivalently, which are Lipschitz on M with respect to the geodesic distance). This was proved by Myers [10] in the case that M is compact, and later on by Nakai [11] in the general case. More recently, analogous results have been obtained in the case of infinite-dimensional Riemannian manifolds (see [12]) and the case of Banach-Finsler manifolds (see [13]). Our aim in this section is to obtain a description of algebra isomorphisms between spaces of type Cb1(M) in the setting of quasi-reversible Finsler manifolds. From this we will obtain a variant of Myers-Nakai Theorem in the context of reversible Finsler manifolds.Now let(M,F) be a Finsler manifold, and let Cb1(M) denote the space of all real bounded C1-smooth functions defined on M whose derivative has uniformly bounded norm. We endow Cb1(M) with the natural norm:
(52)∥f∥Cb1=max{supx∈M|f(x)|,supx∈M∥df(x)∥F}.
Endowed with this norm, Cb1(M) is a complete normed algebra. Note that ∥·∥Cb1 is not submultiplicative, but it satisfies that ∥fg∥Cb1≤2∥f∥Cb1·∥g∥Cb1.Next we are going to recall the definition of the  structure space associated toCb1(M). This construction is standard, but we give some details for the reader’s convenience. Let 𝔐(M) denote the set of all nonzero, multiplicative, continuous linear forms φ:Cb1(M)→ℝ.Claim.
Eachφ∈𝔐(M) satisfies that ∥φ∥=1, and furthermore φ is positive, that is, φ(f)≥0 whenever f≥0.Proof.
Sinceφ is multiplicative and nonzero, it is clear that φ(1)=1. Now we are going to see that, for every f∈Cb1(M), we have that φ(f) belongs to the closure of f(M). Indeed, if α=φ(f) is not in the closure of f(M), then (f-α)2≥ɛ for some ɛ>0. Thus 1/(f-α)2∈Cb1(M) and
(53)1=φ((f-α)2·1(f-α)2)=φ((f-α)2)·φ(1(f-α)2).
But we have that φ((f-α)2)=0, which is a contradiction. From this we obtain that φ is positive. We also obtain that |φ(f)|≤supx∈M|f(x)| for every f∈Cb1(M), so we deduce that ∥φ∥=1.
We endow𝔐(M) with the weak* topology it inherits from the dual space Cb1(M)*. Since 𝔐(M) is a weak*-closed subset of the unit ball, we see that 𝔐(M) is a compact space. Now we consider the embedding δ:M→𝔐(M) given by δ(x)=δx, where δx(f)=f(x) for every f∈Cb1(M) and x∈M. Note that every C1-smooth function f:M→ℝ with compact support belongs to Cb1(M), and thus, in particular, Cb1(M) separates points and closed sets of M. From this it is not difficult to deduce that δ is a  topological embedding, that is, a net (xι) in M converges to x if and only if the net of evaluations (δxι) converges to δx in the weak* topology. On the other hand, the set of evaluations δ(M) is dense in 𝔐(M). Indeed, let φ∈𝔐(M), and consider a weak* basic neighborhood of φ of the form
(54)W={ψ∈𝔐(M):|ψ(fj)-φ(fj)|<ɛ,forj=1,…,m},
where f1,…,fm∈Cb1(M) and ɛ>0. Then there is some x∈M such that δx∈W, since otherwise the function g=∑j=1m‍(fj-φ(fj))2∈Cb1(M) would satisfy g≥ɛ2 and φ(g)=0, and this is impossible since φ is positive. In this way we see that 𝔐(M) is a  compactification of M.In what follows, we concentrate on the case of complete, quasi-reversible Finsler manifolds. Our next lemma will provide a topological characterization of point evaluations inside the structure space.Lemma 14.
Let(M,F) be a connected, second countable, (forward) complete, quasi-reversible Finsler manifold, and let φ∈𝔐(M). The following conditions are equivalent:  (1)
φ has a countable neighborhood basis in 𝔐(M).(2)
There exists somex∈M such that φ=δx.Proof.
(
1
)
⇒
(
2
) Since M is (forward) complete, by Theorem 13, there exists a proper C1-smooth function g:M→ℝ whose differential is uniformly bounded in norm. Suppose now that φ∈𝔐(M)∖δ(M) has a countable neighborhood basis in 𝔐(M). Since δ(M) is dense in 𝔐(M), there is a sequence (xn) in M such that (δxn) converges to φ. Since φ∉δ(M), we see that (xn) has no convergent subsequence in M. Since g is proper, we deduce that limn→∞|g(xn)|=+∞. Then there is a subsequence (xnk) such that |g(xnk+1)|>1+|g(xnk)| for every k. Now we can choose a C1 function θ:ℝ→[0,1], with bounded derivative, such that θ(g(xn2k+1))=1 and θ(g(xn2k))=0 for every k. Then the function f=θ∘g belongs to Cb1(M), but the sequence (δxnk(f)) is not convergent, which is a contradiction.
(
2
)
⇒
(
1
) Conversely, if φ=δx for some x∈M, consider a countable neighborhood basis (Vn) of x in M. Then the family of closures {cl𝔐(M)Vn} is easily seen to be a countable neighborhood basis of δx in 𝔐(M) as required.The following Lemma shows the metric properties of the embeddingδ:M→𝔐(M).Lemma 15.
Let(M,F) be a connected, second countable (forward) complete, quasi-reversible Finsler manifold, with constant C. Then, for each x,y∈M, we have that
(55)1Cmin{1,dF(x,y)}≤∥δx-δy∥≤dF(x,y).Proof.
Recall that(56)∥δx-δy∥=sup{|f(x)-f(y)|:f∈Cb1(M);∥f∥Cb1≤1}.
Thus by the mean value inequality contained in Theorem 5 we deduce at once that ∥δx-δy∥≤dF(x,y). For the other inequality, suppose that x≠y, and consider the function Φ:M→ℝ defined by
(57)Φ(u)=1Cmin{1,dF(x,u)}.
It is clear that 0≤Φ≤1/C≤1, and from Theorem 11 we have that Lip(Φ)≤1. Now given 0<ɛ<(1/2C)min{1,dF(x,y)}, by Theorem 8, there exists a C1-smooth function f:M→ℝ such that |f(u)-Φ(u)|≤ɛ, for every u∈M, and Lip(f)≤1+ɛ. Thus ∥f∥Cb1≤1+ɛ. Now we consider f^=(1/(1+ɛ))f, and we have that ∥f^∥Cb1≤1. Furthermore,
(58)|f^(x)-f^(y)|=11+ɛ|f(x)-f(y)|≥11+ɛ(|Φ(x)-Φ(y)|-2ɛ)=11+ɛ1Cmin{1,dF(x,y)}-2ɛ1+ɛ,
and the result follows.Recall that, ifM and N are Finsler manifolds, a mapping T:Cb1(N)→Cb1(M) is said to be a  normed algebra isomorphism provided T is a bicontinuous linear bijection such that T(f·g)=T(f)·T(g) for every f,g∈Cb1(N). Now we give the main result in this section, which provides a characterization of such normed algebra isomorphisms.Theorem 16.
Let(M,FM) and (N,FN) be second countable, connected, (forward) complete, quasi-reversible Finsler manifolds, with constants CM and CN, respectively. For a mapping T:Cb1(N)→Cb1(M), the following are equivalent:  (1)
T is a normed algebra isomorphism.(2)
There exists aC1-diffeomorphism, h:M→N, which is bi-Lipschitz for the respective Finsler distances, such that T(f)=f∘h, for every f∈Cb1(N).
Moreover, in this case, the bi-Lipschitz constant ofh can be chosen to be
(59)max{CN·∥T∥,CM·∥T-1∥}.Proof.
(
1
)
⇒
(
2
) Suppose that T:Cb1(N)→Cb1(M) is an isomorphism of normed algebras. Consider the transpose map T*:Cb1(M)*→Cb1(N)*, defined by T*(φ)=φ∘T for every φ∈Cb1(M)*. Since T is multiplicative and T* is weak*-to-weak* bicontinuous, we see that the restriction of T* defines a homeomorphism from 𝔐(M) onto 𝔐(N). Consider now the natural embeddings δM:M→𝔐(M) and δN:N→𝔐(N). By Lemma 14 we deduce that T*(δM(M))=δN(N), so that the restriction of T* defines a homeomorphism from δM(M) onto δN(N). Thus we can define h:=(δN)-1∘T*∘δM:M→N, which is a homeomorphism from M onto N. Furthermore, we have that, for every x∈M and f∈Cb1(N),
(60)T(f)(x)=δx(T(f))=T*(δx)(f)=δh(x)(f)=f(h(x)),
that is, T(f)=f∘h. In particular, note that f∘h is C1-smooth for every C1-smooth function f:N→ℝ with compact support. From this it is easily deduced that h is C1-smooth, and the same can be said about h-1, so that h is a C1-diffeomorphism.
Now we are going to see thath:M→N is bi-Lipschitz for the respective Finsler distances dM and dN. Using Proposition 6, it will suffice to prove that h and h-1 are locally Lipschitz. Given p∈M, consider the open neighborhood
(61)Up=Bp+(12)∩Bp-(12)∩h-1(Bh(p)+(12))∩h-1(Bh(p)-(12)).
If x,y∈Up we have that dM(x,y)<1 and dN(h(x),h(y))<1, so by Lemma 15 and taking into account that ∥T*∥=∥T∥, we obtain that
(62)1CNdN(h(x),h(y))≤∥δh(x)-δh(y)∥=∥T*(δx-δy)∥≤∥T∥·∥δx-δy∥≤∥T∥dM(x,y).
Therefore, we have that h is CN·∥T∥-Lipschitz. In the same way, h-1 is CM·∥T-1∥-Lipschitz and the result follows.
(
2
)
⇒
(
1
) Taking into account Theorem 5, we have that Cb1(M) is the set of all C1-smooth functions f:M→ℝ which are Lipschitz, and the same holds for Cb1(N), so this implication is clear.We are now ready to deduce from our previous results a version of the classical Myers-Nakai Theorem in the context of reversible Finsler manifolds. Recall that a mappingh:(M,FM)→(N,FN) between Finsler manifolds is said to be a  Finsler isometry if h is a C1 diffeomorphism which preserves the Finsler structure, that is, for every x∈M and every v∈TxM,
(63)FM(x,v)=FN(h(x),dh(x)(v)).Extending the classical result by Myers and Steenrod [14] about Riemannian manifolds, Deng and Hou have proved in ([7], Theorem 2.2) that, if (M,FM) and (N,FN) are connected Finsler manifolds, a mapping h:M→N is a Finsler isometry if and only if h is bijective and preserves the corresponding Finsler distances dM and dN, that is, for every x,y∈M,
(64)dM(x,y)=dN(h(x),h(y)).Now combining the above result with Theorem16 we obtain at once the following theorem, which has been also obtained in [4] (see Theorem 13 there) but with a different proof.Theorem 17.
LetM and N be second countable, connected, reversible, and complete Finsler manifolds. Then M and N are equivalent as Finsler manifolds if and only if Cb1(M) and Cb1(N) are isometric normed algebras. Moreover, every normed algebra isometry T:Cb1(N)→Cb1(M) is of the form T(f)=f∘h, where h:M→N is a Finsler isometry.



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*Source: 164571-2013-07-29.xml*