Smooth Approximation of Lipschitz Functions on Finsler Manifolds

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 function 𝑓∶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 𝐶¹_𝑏(𝑀) of all 𝐶¹ functions with bounded derivative on a complete quasi-reversible Finsler manifold 𝑀, we obtain a characterization of algebra isomorphisms 𝑇∶𝐶¹_𝑏(𝑁)→𝐶¹_𝑏(𝑀) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

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 Section 2 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 𝐶¹_𝑏(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∶𝐶¹_𝑏(𝑁)→𝐶¹_𝑏(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 norm

Definition 1. Let 𝑉 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 every 𝑣∈𝑉⧵{0}, the quadratic form 𝑔_𝑣 associated to the second derivative of the function 𝐹² at 𝑣, that is, 𝑔_𝑣=12𝑑²𝐹²(𝑣), (1) is positive definite on 𝑉.

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 𝐹(𝜆𝑣)=|𝜆|𝐹(𝑣),forevery𝑣∈𝑉andevery𝜆∈ℝ. (2) In this case, 𝐹 is a norm in the usual sense.

Now the definition of Finsler manifold is as follows.

Definition 2. A Finsler 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 every 𝑥∈𝑀, 𝐹(𝑥,⋅)∶𝑇_𝑥𝑀→[0,∞) is a Minkowski norm on the tangent space 𝑇_𝑥𝑀.

In particular, a Riemannian manifold is a special case of Finsler manifold, where the Minkowski norm on each tangent space 𝑇_𝑥𝑀 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  𝑑_𝐹 on 𝑀 is defined by 𝑑_𝐹𝑥,𝑦=infℓ_𝐹(𝜎)∶𝜎piecewise𝐶¹pathfrom𝑥to𝑦, (3) where the Finsler length of a piecewise 𝐶¹ path 𝜎∶[𝑎,𝑏]→𝑀 is defined as: ℓ_𝐹(𝜎)=^𝑏_𝑎𝐹𝜎(𝑡),𝜎^′(𝑡)𝑑𝑡. (4) In this way we have (see Section 6.2 of [6]) that the Finsler distance 𝑑_𝐹 is the so-called an asymmetric distance on 𝑀, in the sense that it verifies (i)𝑑_𝐹(𝑥,𝑦)≥0.(ii)𝑑_𝐹(𝑥,𝑦)=0 if and only if 𝑥=𝑦.(iii)𝑑_𝐹(𝑥,𝑦)≤𝑑_𝐹(𝑥,𝑧)+𝑑_𝐹(𝑧,𝑦), for every 𝑥,𝑦,𝑧∈𝑀.

In general, 𝑑_𝐹 needs not to be symmetric. Nevertheless, when 𝐹 is reversible the Finsler distance 𝑑_𝐹 is symmetric, and therefore (𝑀,𝑑_𝐹) 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 𝐁⁺_𝑝(𝑟)=𝑥∈𝑀∶𝑑_𝐹𝑝,𝑥<𝑟. (5) In the same way, the backward ball of center 𝑝 and radius 𝑟 is defined as 𝐁⁻_𝑝(𝑟)=𝑥∈𝑀∶𝑑_𝐹𝑥,𝑝<𝑟. (6) 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 𝑑_𝐹 is given by 𝑑_𝐹(𝑢,𝑣)=𝐹(𝑣−𝑢). (7) 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, 𝐵₀(𝑟)=𝐁⁺₀(𝑟)={𝑣∈𝑉∶𝐹(𝑣)<𝑟}. (8)

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 exp_𝑥∶𝐵₀(𝑟)→𝐁⁺_𝑥(𝑟) is a 𝐶¹-diffeomorphism. Then, for 𝑢,𝑣∈𝐵₀(𝑟) with 𝑢≠𝑣, one has that lim_{(𝑢,𝑣)→(0,0)}𝐹(𝑥,𝑣−𝑢)𝑑_𝐹exp_𝑥(𝑢),exp_𝑥(𝑣)=1. (9)

A terminological remark is now in order. Suppose that 𝑋 and 𝑌 are two nonempty sets, endowed with asymmetric distances 𝑑_𝑋 and 𝑑_𝑌, respectively. We say that a mapping 𝑓∶(𝑋,𝑑_𝑋)→(𝑌,𝑑_𝑌) is Lipschitz, with constant 𝐶>0 (or, briefly, 𝐶-Lipschitz) if, for every 𝑥₁,𝑥₂∈𝑋, 𝑑_𝑌𝑓𝑥₁,𝑓𝑥₂≤𝐶⋅𝑑_𝑋𝑥₁,𝑥₂. (10) As usual, we will say that 𝑓∶(𝑋,𝑑_𝑋)→(𝑌,𝑑_𝑌) 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 𝜀>0, there exists 𝑟>0 such that the exponential mapping exp_𝑥∶𝐵₀(𝑟)→𝐁⁺_𝑥(𝑟) is a 𝐶¹-diffeomorphism, which is (1+𝜀)-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 𝑓𝑓(𝑥)𝑦Lip=sup−𝑓𝑑_𝐹∈[]𝑥,𝑦∶𝑥,𝑦∈𝑀,𝑥≠𝑦0,∞. (11) 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 𝑑𝑓(𝑥)_𝐹=sup𝑑𝑓(𝑥)(𝑣)∶𝑣∈𝑇_𝑥𝑀,𝐹(𝑥,𝑣)≤1. (12) 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, Lip𝑓=sup𝑑𝑓(𝑥)_𝐹∶𝑥∈𝑀. (13) Thus, for every 𝑝,𝑞∈𝑀, one has that 𝑓𝑝−𝑓𝑞≤sup𝑑𝑓(𝑥)_𝐹∶𝑥∈𝑀⋅𝑑_𝐹𝑝,𝑞. (14)

Proof. For the proof, fix a number 𝐶≥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∈𝑇_𝑝𝑀 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)=exp_𝑝(𝑡𝑣), defined for 𝑡∈[0,𝑟], minimizes the Finsler distance from point 𝑝, that is, 𝑑_𝐹(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 ℎ(𝑡)−ℎ(0)𝑡>𝐶,if0<𝑡≤𝛿. (15) Since 𝑝=𝛾(0), choosing 𝑞=𝛾(𝛿), we obtain that 𝑓𝑝−𝑓𝑞=ℎ(𝛿)−ℎ(0)>𝐶⋅𝛿=𝐶⋅𝑑_𝐹𝑝,𝑞 (16) which contradicts the fact that 𝑓 is 𝐶-Lipschitz.
(2)⇒(1) Consider 𝑥,𝑦∈𝑀, and let 𝜀>0. Choose a piecewise 𝐶¹ path 𝛾∶[𝑎,𝑏]→𝑀 such that 𝛾(𝑎)=𝑥, 𝛾(𝑏)=𝑦, and ℓ_𝐹𝛾≤𝑑_𝐹𝑥,𝑦+𝜀. (17) Define now ℎ∶[𝑎,𝑏]→ℝ by ℎ(𝑡)=𝑓(𝛾(𝑡)). Then, 𝑓𝑦−𝑓(𝑥)=^𝑏_𝑎ℎ^′(𝑡)𝑑𝑡≤^𝑏_𝑎ℎ^′(𝑡=)𝑑𝑡_𝑏𝑎𝑑𝑓𝛾(𝑡)𝛾^′(𝑡)≤𝑑𝑡_𝑏𝑎𝑑𝑓𝛾(𝑡)_𝐹⋅𝐹𝛾(𝑡),𝛾^′(𝑡)𝑑𝑡≤𝐶^𝑏_𝑎𝐹𝛾(𝑡),𝛾^′(𝑡)𝑑𝑡=𝐶⋅ℓ_𝐹𝛾≤𝐶𝑑_𝐹𝑥,𝑦+𝜀. (18) This shows that |𝑓(𝑥)−𝑓(𝑦)|≤𝐶⋅𝑑_𝐹(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,𝐹_𝑀) and (𝑁,𝐹_𝑁) be connected Finsler manifolds, with Finsler distances 𝑑_𝑀 and 𝑑_𝑁, respectively. A mapping 𝑓∶(𝑀,𝑑_𝑀)→(𝑁,𝑑_𝑁) is 𝐶-Lipschitz if and only if it is locally 𝐶-Lipschitz; that is, every point 𝑝∈𝑀 has a neighborhood 𝑈^𝑝 such that, for every 𝑥,𝑦∈𝑈^𝑝, 𝑑_𝑁𝑓(𝑥),𝑓𝑦≤𝐶⋅𝑑_𝑀𝑥,𝑦. (19)

Proof. Suppose that 𝑓 is locally 𝐶-Lipschitz. Consider 𝑥,𝑦∈𝑀 and 𝜀>0. Choose a piecewise 𝐶¹ path 𝛾∶[𝑎,𝑏]→𝑀 from 𝑥 to 𝑦, such that ℓ_𝑀(𝛾)≤𝑑_𝑀(𝑥,𝑦)+𝜀. Each 𝑝∈𝛾([𝑎,𝑏]) has an open neighborhood 𝑈^𝑝 where 𝑓 is 𝐶-Lipschitz. Choose a partition 𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑘=𝑏 of [𝑎,𝑏] such that, for every 𝑖=1,…,𝑘,  𝛾([𝑡_𝑖−1,𝑡_𝑖]) is contained into 𝑈^𝑝 for some 𝑝∈𝛾([𝑎,𝑏]). Then 𝑑_𝑁𝑓(𝑥),𝑓𝑦≤^𝑘_𝑖=1𝑑_𝑁𝑓𝛾𝑡_𝑖−1,𝑓𝛾𝑡_𝑖≤𝐶⋅^𝑘_𝑖=1𝑑_𝑀𝛾𝑡_𝑖−1,𝛾𝑡_𝑖≤𝐶⋅^𝑘_𝑖=1ℓ_𝑀𝛾_{[𝑡𝑖−1,𝑡𝑖]}=𝐶⋅ℓ_𝑀𝛾≤𝐶⋅𝑑_𝑀𝑥,𝑦+𝜀. (20) 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 Theorem 8 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 𝜀>0 is given, there exists a 𝐶¹-smooth Lipschitz function 𝑔∶𝑀→ℝ such that |𝑓(𝑥)−𝑔(𝑥)|≤𝜀, for every 𝑥∈𝑀, and Lip(𝑔)≤Lip(𝑓)+𝜀. 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 𝐴_𝛿={𝑣+𝑤∶𝑣∈𝐴,𝐹(𝑤)<𝛿}=_𝑣∈𝐴𝐵_𝑣(𝛿). (21) Suppose that 𝑓∶𝐴_𝛿→ℝ is Lipschitz and let 𝜀>0. Then there exists a 𝐶^∞-smooth function 𝑔∶𝑉→ℝ such that |𝑓(𝑣)−𝑔(𝑣)|≤𝜀, for every 𝑣∈𝐴, and Lip(𝑔|_𝐴)≤Lip(𝑓|_𝐴).

Proof. By choosing a basis of 𝑉, we may assume that 𝑉=ℝ^𝑛. Note that, by local compactness, it follows that the Minkowski norm 𝐹 is equivalent to the usual Euclidean norm ‖⋅‖ in ℝ^𝑛, in the sense that there exists some 𝑅≥1 such that 1𝑅⋅‖𝑣‖≤𝐹(𝑣)≤𝑅⋅‖𝑣‖ (22) for every 𝑣∈ℝ^𝑛. Now, if 𝑓∶𝐴_𝛿→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∶ℝ^𝑛→ℝ.
Now consider a sequence (𝜑_𝑘)_𝑘 of usual 𝐶^∞-smooth mollifiers on ℝ^𝑛, where each 𝜑_𝑘 is nonnegative, supp(𝜑_𝑘) is contained in the Euclidean ball 𝔹(0,1/𝑘), and ∫_ℝ𝑛𝜑_𝑘=1. For each 𝑘, define 𝑓_𝑘∶ℝ^𝑛→ℝ by 𝑓_𝑘(𝑣)=_ℝ𝑛𝑓(𝑣+𝑤)𝜑_𝑘(𝑤)𝑑𝑤. (23) Each 𝑓_𝑘 is 𝐶^∞-smooth, and, since 𝑓 is uniformly continuous, we have that the sequence (𝑓_𝑘) converges to 𝑓 uniformly on ℝ^𝑛. Given 𝜀>0, choose 𝑘>𝑅/𝛿 and large enough so that ‖𝑓_𝑘−𝑓‖_∞<𝜀 and define 𝑔=𝑓_𝑘. Then, if 𝑢,𝑣∈𝐴, 𝑔(𝑢)−𝑔(𝑣)=_ℝ𝑛𝑓(𝑢+𝑤)𝜑_𝑘(𝑤)−𝑑𝑤_ℝ𝑛𝑓(𝑣+𝑤)𝜑_𝑘(𝑤)𝑑𝑤≤_ℝ𝑛𝑓(𝑢+𝑤)−𝑓(𝑣+𝑤)⋅𝜑_𝑘(𝑤)𝑑𝑤=_{𝔹(0,1/𝑘)}𝑓(𝑢+𝑤)−𝑓(𝑣+𝑤)⋅𝜑_𝑘(𝑤()⋅)𝑑𝑤≤𝐶⋅𝐹𝑣−𝑢_ℝ𝑛𝜑_𝑘(𝑤)𝑑𝑤=𝐶⋅𝑑_𝐹(𝑢,𝑣). (24) 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 𝜀∶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)|≤𝜀(𝑥), for every 𝑥∈𝑀, and Lip(𝑔)≤Lip(𝑓)+𝑟.

Proof. Let us denote 𝐶=Lip(𝑓). Without loss of generality we may assume that, for every 𝑥∈𝑀, 𝜀(𝑥)>0 is small enough so that 𝜀(𝑥)≤𝑟/2 and 𝐶⋅(1+𝜀(𝑥))²𝑟<𝐶+2. (25) Using Corollary 4, for each 𝑥∈𝑀, we can choose 𝛿_𝑥>0 such that the exponential mapping exp_𝑥 is a 𝐶¹-diffeomorphism and (1+𝜀(𝑥))-bi-Lipschitz from the Minkowski ball 𝐵_0𝑥(3𝛿_𝑥)⊂𝑇_𝑥𝑀 onto the forward ball 𝐁⁺_𝑥(3𝛿_𝑥)⊂𝑀, where 0_𝑥 denotes the null vector of 𝑇_𝑥𝑀. In addition, by the continuity of 𝑓 and 𝜀, we can also assume that 𝜀(𝑦)≥𝜀(𝑥)/2 and |𝑓(𝑦)−𝑓(𝑥)|≤𝜀(𝑥)/2, for every 𝑦∈𝐁⁺_𝑥(3𝛿_𝑥). Since 𝑀 is second countable, there is a sequence (𝑥_𝑛) in 𝑀 such that 𝑀=^∞_𝑛=1𝐁⁺_𝑥𝑛𝛿_𝑛, (26) where we denote 𝛿_𝑛=𝛿_𝑥𝑛. Now, for each 𝑛∈ℕ, we define 𝑓_𝑛∶𝐵_0𝑥𝑛(3𝛿_𝑛)→ℝ by 𝑓_𝑛(𝑣)=𝑓exp_𝑥𝑛(𝑣), (27) and we then have that 𝑓_𝑛 is 𝐶⋅(1+𝜀(𝑥_𝑛))-Lipschitz.
Next, we are going to construct a partition of unity subordinated to the covering {𝐁⁺_𝑥𝑛(2𝛿_𝑛)}_𝑛∈ℕ of 𝑀, estimating the Lipschitz constant of the respective functions. Thus, for each 𝑛∈ℕ, let 𝜃_𝑛∶ℝ→[0,1] be a 𝐶^∞-smooth function such that 𝜃_𝑛(𝑥)=1if𝑥∈−∞,𝛿_𝑛,0if𝑥∈2𝛿_𝑛,+∞, (28) and define 𝜑_𝑛∶𝑀→[0,1] by 𝜑_𝑛(𝑥)=𝜃_𝑛𝐹𝑥_𝑛,exp_𝑥⁻¹_𝑛(𝑥)if𝑥∈𝐁⁺_𝑥𝑛3𝛿_𝑛,0otherwise. (29) It is clear that each 𝜑_𝑛 is 𝐶¹-smooth and Lipschitz. Furthermore, 𝜑_𝑛=1 on the forward ball 𝐁⁺_𝑥𝑛(𝛿_𝑛) and 𝜑_𝑛=0 on 𝑀⧵𝐁⁺_𝑥𝑛(2𝛿_𝑛). Now we define the functions 𝜓_𝑛∶𝑀→[0,1] by setting 𝜓₁=𝜑₁ and, for 𝑛≥2, 𝜓_𝑛=𝜑_𝑛⋅_𝑗<𝑛1−𝜑_𝑗. (30) Then, it is easy to check that, for every 𝑛∈ℕ, (i)𝜓_𝑛 is 𝐶¹-smooth and 𝐶_𝑛-Lipschitz, where 𝐶_𝑛=∑_𝑗≤𝑛Lip(𝜑_𝑗). (ii)supp(𝜓_𝑛)⊂supp(𝜑_𝑛)⊂𝐁⁺_𝑥𝑛(2𝛿_𝑛). (iii)𝜓_𝑚=0 on 𝐁⁺_𝑥𝑛(𝛿_𝑛), whenever 𝑚>𝑛.
Thus, {𝜓_𝑛} is the desired partition of unity. Indeed, for each 𝑥∈𝑀, let 𝑛=𝑛(𝑥) be the first integer such that 𝑥∈𝐁⁺_𝑥𝑛(𝛿_𝑛). Then 𝜑_𝑛(𝑥)=1 and 𝜓_𝑚(𝐁⁺_𝑥𝑛(𝛿_𝑛))=0, for 𝑚>𝑛. Therefore, the family {supp(𝜓_𝑚)}_𝑚 is locally finite. In addition ∑_𝑚𝜓_𝑚(𝑥)=1, since _𝑚𝜓_𝑚(𝑥)=𝜓₁(𝑥)+𝜓₂(𝑥)+⋯+𝜓_𝑛(𝑥)=𝜓₁(𝑥)+𝜓₂(𝑥)+⋯+𝜓_𝑛−1(𝑥)+𝜑_𝑛(𝑥)_𝑗<𝑛1−𝜑_𝑗(𝑥)=𝜓₁(𝑥)+𝜓₂(𝑥)+⋯+𝜑_𝑛−1(𝑥)×_{𝑗<𝑛−1}1−𝜑_𝑗(𝑥)+_𝑗<𝑛1−𝜑_𝑗(𝑥)=𝜓₁(𝑥)+𝜓₂(𝑥)+⋯+𝜑_𝑛−2(𝑥)×_{𝑗<𝑛−2}1−𝜑_𝑗(𝑥)+_{𝑗<𝑛−1}1−𝜑_𝑗(𝑥)=⋯=𝜑₁(𝑥)+1−𝜑₁(𝑥)=1. (31) Now using Lemma 7 we can find, for each 𝑛, a 𝐶^∞-smooth function 𝑔_𝑛∶𝑇_𝑥𝑛𝑀→ℝ such that 𝑔_𝑛(𝑣)−𝑓_𝑛(𝑣≤𝜀𝑥)^𝑛2^𝑛+2𝐶_𝑛+1 (32) for every 𝑣∈𝐵_0𝑥𝑛(2𝛿_𝑛), and Lip𝑔_{𝑛𝐵0𝑥𝑛(2𝛿𝑛)}≤Lip𝑓_{𝑛𝐵0𝑥𝑛(2𝛿𝑛)}≤𝐶⋅1+𝜀𝑥_𝑛. (33) Thus, we define the approximation function 𝑔∶𝑀→ℝ by 𝑔(𝑥)=_𝑛𝜓_𝑛(𝑥)⋅𝑔_𝑛exp_𝑥⁻¹_𝑛(𝑥) (34) for each 𝑥∈𝑀. Note that, since the exponential mapping exp_𝑥𝑛 is a 𝐶¹-diffeomorphism from 𝐵_0𝑥𝑛(3𝛿_𝑛) onto 𝐁⁺_𝑥𝑛(3𝛿_𝑛), the expression 𝜓_𝑛(𝑥)⋅𝑔_𝑛(exp_𝑥⁻¹_𝑛(𝑥)) is well defined for 𝑥∈𝐁⁺_𝑥𝑛(3𝛿_𝑛) and it is 𝐶¹-smooth on 𝐁⁺_𝑥𝑛(3𝛿_𝑛). On the other hand, if 𝑥∉𝐁⁺_𝑥𝑛(2𝛿_𝑛)⊃supp(𝜓_𝑛), then 𝜓_𝑛(𝑥)=0. Thus, for every 𝑥∉𝐁⁺_𝑥𝑛(3𝛿_𝑛), we may assume that 𝜓_𝑛(𝑥)⋅𝑔_𝑛(exp_𝑥⁻¹_𝑛(𝑥)) is zero. With this convention, and taking into account that {𝜓_𝑛} is a 𝐶¹-smooth partition of unity, we obtain that 𝑔 is well defined and 𝐶¹-smooth on 𝑀.
We are going to see that 𝑔 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∈𝐁⁺_𝑥𝑛(𝛿_𝑛). To simplify, denote 𝑣_𝑚=exp_𝑥⁻¹_𝑚(𝑥)∈𝑇_𝑥𝑚𝑀, for all 𝑚. Then we have 𝑔(𝑥)−𝑓(𝑥)=_𝑚≤𝑛𝜓_𝑚(𝑥)⋅𝑔_𝑚exp_𝑥⁻¹_𝑚(𝑥)(𝑥=−𝑓)_𝑚≤𝑛𝜓_𝑚(𝑥)⋅𝑔_𝑚𝑣_𝑚(𝑥)=−𝑓_𝑚≤𝑛𝜓_𝑚(𝑥)⋅𝑔_𝑚𝑣_𝑚−𝑓_𝑚𝑣_𝑚≤_𝑚≤𝑛𝜓_𝑚(𝑥)𝜀𝑥_𝑚2^𝑚+2𝐶_𝑚+1≤_𝑚≤𝑛𝜓_𝑚(𝑥)𝜀𝑥_𝑚2≤_𝑚≤𝑛𝜓_𝑚(𝑥)𝜀(𝑥)≤𝜀(𝑥). (35) 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∈𝐁⁺_𝑥𝑛(𝛿_𝑛). We are going to see that 𝑔 is (𝐶+𝑟)-Lipschitz on the open set 𝑈^𝑧=𝐁⁺_𝑧(𝛿_𝑧)∩𝐁⁻_𝑧(𝛿_𝑧), where 𝛿_𝑧=12min𝛿₁,…,𝛿_𝑛,𝛿_𝑛−𝑑_𝐹𝑥_𝑛,𝑧. (36) For each 𝑥,𝑦∈𝑀, denote 𝑃_𝑥,𝑦=𝑚∈{1,…,𝑛}∶𝐁⁺_𝑥𝑚2𝛿_𝑚∩𝑥,𝑦≠∅. (37) Then, whenever 𝑥,𝑦∈𝑈^𝑧, the following holds:(1) if 𝑚∈{1,…,𝑛} and 𝑦∈𝐁⁺_𝑥𝑚(2𝛿_𝑚), then 𝑥∈𝐁⁺_𝑥𝑚(3𝛿_𝑚). Indeed, if 𝑥,𝑦∈𝑈^𝑧, then 𝑑_𝐹(𝑦,𝑥)<2𝛿_𝑧≤𝛿_𝑚. Thus if 𝑦∈𝐁⁺_𝑥𝑚(2𝛿_𝑚) we have that 𝑑_𝐹𝑥_𝑚,𝑥≤𝑑_𝐹𝑥_𝑚,𝑦+𝑑_𝐹𝑦,𝑥<2𝛿_𝑚+𝛿_𝑚=3𝛿_𝑚, (38) and therefore 𝑥∈𝐁⁺_𝑥𝑚(3𝛿_𝑚).(2) For every 𝑚∈𝑃_𝑥,𝑦 we have that 𝑥,𝑦∈𝐁⁺_𝑥𝑚(3𝛿_𝑚). That is clear from the above. In particular, if 𝑚∈𝑃_𝑥,𝑦 then 𝑣_𝑚=exp_𝑥⁻¹_𝑚(𝑥) and 𝑤_𝑚=exp_𝑥⁻¹_𝑚(𝑦) are well defined, and, using (33) and the fact that exp_𝑥⁻¹_𝑚∶𝐁⁺_𝑥𝑚(3𝛿_𝑚)→𝐵_0𝑥𝑚(3𝛿_𝑚) is (1+𝜀(𝑥_𝑚))-Lipschitz, we see that 𝑔_𝑚𝑣_𝑚−𝑔_𝑚𝑤_𝑚≤𝐶⋅1+𝜀𝑥_𝑚×𝑑_𝐹exp_𝑥⁻¹_𝑚(𝑥),exp_𝑥⁻¹_𝑚𝑦≤𝐶⋅1+𝜀𝑥_𝑚²𝑑_𝐹𝑥,𝑦. (39)(3) If 𝑚∈ℕ⧵𝑃_𝑥,𝑦 then 𝜓_𝑚(𝑥)=0=𝜓_𝑚(𝑦). That follows, since supp(𝜓_𝑚)⊂𝐁⁺_𝑥𝑚(2𝛿_𝑚) and supp(𝜓_ℓ)∩𝐁⁺_𝑥𝑛(𝛿_𝑛)=∅, for every 𝑙>𝑛.
As a consequence of the above we have, for every 𝑥,𝑦∈𝑈^𝑧, and using the notation 𝑣_𝑚=exp_𝑥⁻¹_𝑚(𝑥) and 𝑤_𝑚=exp_𝑥⁻¹_𝑚(𝑦), the following: (i)𝑔(𝑥)=∑_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑥)⋅𝑔_𝑚(𝑣_𝑚), (ii)𝑔(𝑦)=∑_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑦)⋅𝑔_𝑚(𝑤_𝑚), (iii)∑_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑥)=1=∑_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑦), (iv)|𝑔_𝑚(𝑣_𝑚)−𝑔_𝑚(𝑤_𝑚)|≤𝐶(1+𝜀(𝑥_𝑚))²𝑑_𝐹(𝑥,𝑦), whenever 𝑚∈𝑃_𝑥,𝑦.
Therefore, since _{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑥)−𝜓_𝑚𝑦⋅𝑓(𝑥)=0, (40) we deduce that 𝑔(𝑥)−𝑔𝑦=_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑥)⋅𝑔_𝑚𝑣_𝑚−_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚𝑦⋅𝑔_𝑚𝑤_𝑚=_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑥)−𝜓_𝑚𝑦⋅𝑔_𝑚𝑣_𝑚−𝑓(𝑥)+_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚𝑦⋅𝑔_𝑚𝑣_𝑚−𝑔_𝑚𝑤_𝑚. (41) Finally, using (25), (32), and the fact that 𝜓_𝑚 is 𝐶_𝑚-Lipschitz, we then have that 𝑔(𝑥)−𝑔𝑦≤_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚(𝑥)−𝜓_𝑚𝑦⋅𝑔_𝑚𝑣_𝑚(𝑥+−𝑓)_{𝑚∈𝑃𝑥,𝑦}𝜓_𝑚𝑦⋅𝑔_𝑚𝑣_𝑚−𝑔_𝑚𝑤_𝑚≤_𝑚≤𝑛𝐶_𝑚⋅𝜀𝑥_𝑚𝐶_𝑚+12^𝑚+2𝑑_𝐹𝑥,𝑦+_𝑚≤𝑛𝜓_𝑚𝑦𝑟𝐶+2𝑑_𝐹𝑥,𝑦≤_𝑚≤𝑛𝜀(𝑧)2^𝑚+1𝑑_𝐹+𝑟𝑥,𝑦𝐶+2𝑑_𝐹≤()𝑥,𝑦𝐶+𝑟⋅𝑑_𝐹𝑥,𝑦 (42) since ∑_𝑚≤𝑛(𝜀(𝑧)/2^𝑚+1)≤𝜀(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 only 𝐶¹-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 𝐹(𝑥,−𝑣)≤𝐶⋅𝐹(𝑥,𝑣),forevery(𝑥,𝑣)∈𝑇𝑀. (43)

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 for 𝐶=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 (𝑇_𝑥𝑀,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)⋅𝑑_𝐹(𝑦,𝑥)≤𝑑_𝐹(𝑥,𝑦)≤𝐶⋅𝑑_𝐹(𝑦,𝑥), for every 𝑥,𝑦∈𝑀.(3)For all 𝑝∈𝑀, the forward distance function Φ_𝑝=𝑑_𝐹(𝑝,⋅) is 𝐶-Lipschitz.(4)For all 𝑝∈𝑀, the backward distance function Ψ_𝑝=𝑑_𝐹(⋅,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, ℓ_𝐹𝛾=¹₀𝐹𝛾(𝑡),𝛾^′(𝑡)=𝑑𝑡₁₀𝐹𝛾(1−𝑡),−𝛾^′(=1−𝑡)𝑑𝑡₁₀𝐹𝛾(𝑠),−𝛾^′(𝑠)𝑑𝑠≤𝐶⋅¹₀𝐹𝛾(𝑠),𝛾^′(𝑠)𝑑𝑠=𝐶⋅ℓ_𝐹𝛾. (44) This implies that 𝑑_𝐹(𝑥,𝑦)≤𝐶⋅𝑑_𝐹(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 lim_𝑡→0+𝐹(𝑥,𝑡𝑣−𝑡(−𝑣))𝑑_𝐹exp_𝑥(𝑡(−𝑣)),exp_𝑥(𝑡𝑣)=1=lim_𝑡→0+𝐹(𝑥,𝑡(−𝑣)−𝑡𝑣)𝑑_𝐹exp_𝑥(𝑡𝑣),exp_𝑥(𝑡(.−𝑣)) (45) That is, lim_𝑡→0+𝐹(𝑥,2𝑡𝑣)𝑑_𝐹exp_𝑥(−𝑡𝑣),exp_𝑥(𝑡𝑣)=1=lim_𝑡→0+𝐹(𝑥,2𝑡(−𝑣))𝑑_𝐹exp_𝑥(𝑡𝑣),exp_𝑥(.−𝑡𝑣) (46) Thus we obtain that 𝐹(𝑥,−𝑣)=lim_𝑡→0+𝑑_𝐹exp_𝑥(𝑡𝑣),exp_𝑥(−𝑡𝑣)2𝑡≤𝐶lim_𝑡→0+𝑑_𝐹exp_𝑥(−𝑡𝑣),exp_𝑥(𝑡𝑣)().2𝑡=𝐶⋅𝐹𝑥,𝑣 (47)(2)⇒(3) For every 𝑝,𝑥,𝑦∈𝑀 we have, from the triangle inequality, that 𝑑_𝐹𝑝,𝑥−𝑑_𝐹𝑝,𝑦≤𝑑_𝐹𝑑𝑦,𝑥_𝐹𝑝,𝑦−𝑑_𝐹𝑝,𝑥≤𝑑_𝐹𝑥,𝑦. (48) By the hypothesis (2), we follow at once that 𝑑_𝐹𝑝,𝑥−𝑑_𝐹𝑝,𝑦≤𝐶⋅𝑑_𝐹𝑥,𝑦. (49) This means that the function Φ_𝑝=𝑑_𝐹(𝑝,⋅) is 𝐶-Lipschitz.
(3)⇒(2) For every 𝑝,𝑥,𝑦∈𝑀 we have that, in particular, 𝑑_𝐹𝑝,𝑥−𝑑_𝐹𝑝,𝑦≤𝐶⋅𝑑_𝐹𝑥,𝑦. (50) Choosing 𝑝=𝑦 we have that 𝑑_𝐹𝑦,𝑥≤𝐶⋅𝑑_𝐹𝑥,𝑦. (51) Reversing the roles of 𝑥 and 𝑦, we also have that 𝑑_𝐹(𝑥,𝑦)≥(1/𝐶)⋅𝑑_𝐹(𝑦,𝑥).
(4)⇔(2) This can be seen as before.

Note that, by choosing 𝐶=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)𝑑_𝐹(𝑦,𝑥)=𝑑_𝐹(𝑥,𝑦), for every x,𝑦∈𝑀.

As an application of Theorem 8, 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 (𝑥_𝑛) in a Finsler manifold (𝑀,𝐹) is said to be forward Cauchy (respectively, backward Cauchy) if, for every 𝜀>0, there exists some 𝑛₀∈ℕ such that, if 𝑛₀≤𝑛≤𝑚, then 𝑑_𝐹(𝑥_𝑛,𝑥_𝑚)<𝜀, (respectively, 𝑑_𝐹(𝑥_𝑚,𝑥_𝑛)<𝜀). 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 function 𝑓∶𝑀→ℝ. (3)There exists a proper 𝐶¹-smooth function 𝑔∶𝑀→ℝ whose differential is uniformly bounded in norm.

Proof. (1)⇒(2) Fix 𝑝∈𝑀 and consider the forward distance function 𝜑_𝑝=𝑑_𝐹(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 𝜑_𝑝 is a proper function. Furthermore, by Theorem 11, we have that 𝜑_𝑝 is Lipschitz.
(2)⇒(3) Suppose that the proper function 𝑓∶𝑀→ℝ is 𝐶-Lipschitz, and fix some 𝜀>0. By Theorem 8, there exists a 𝐶¹-smooth function 𝑔∶𝑀→ℝ such that (i)|𝑓(𝑥)−𝑔(𝑥)|≤𝜀, for every 𝑥∈𝑀. (ii)Lip(𝑔)≤𝐶+𝜀.
It is easy to check that 𝑔 is a proper function, since 𝑓 is so. On the other hand, by Theorem 5, we have that ‖𝑑𝑔(𝑥)‖_𝐹≤𝐶+𝜀, for every 𝑥∈𝑀.
(3)⇒(1) Let (𝑥_𝑛) be a forward Cauchy sequence in 𝑀. Since 𝑔∶𝑀→ℝ is Lipschitz, then {𝑔(𝑥_𝑛)} is a Cauchy sequence in ℝ, and therefore {𝑔(𝑥_𝑛)} converges to some point 𝑧 in ℝ. Thus 𝐾={𝑧}∪{𝑔(𝑥_𝑛)∶𝑛∈ℕ} is a compact subset of ℝ. Now (𝑥_𝑛) is contained in 𝑔⁻¹(𝐾), which is compact since 𝑔 is proper. Then (𝑥_𝑛) 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 manifold 𝑀 is determined by the natural normed algebra structure on the space 𝐶¹_𝑏(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 𝐶¹_𝑏(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 𝐶¹_𝑏(𝑀) denote the space of all real bounded 𝐶¹-smooth functions defined on 𝑀 whose derivative has uniformly bounded norm. We endow 𝐶¹_𝑏(𝑀) with the natural norm: 𝑓_𝐶1𝑏=maxsup_𝑥∈𝑀𝑓(𝑥),sup_𝑥∈𝑀𝑑𝑓(𝑥)_𝐹. (52) Endowed with this norm, 𝐶¹_𝑏(𝑀) is a complete normed algebra. Note that ‖⋅‖_𝐶¹_𝑏 is not submultiplicative, but it satisfies that ‖𝑓𝑔‖_𝐶1𝑏≤2‖𝑓‖_𝐶1𝑏⋅‖𝑔‖_𝐶1𝑏.

Next we are going to recall the definition of the structure space associated to 𝐶¹_𝑏(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∶𝐶¹_𝑏(𝑀)→ℝ.

Claim. Each 𝜑∈𝔐(𝑀) satisfies that ‖𝜑‖=1, and furthermore 𝜑 is positive, that is, 𝜑(𝑓)≥0 whenever 𝑓≥0.

Proof. Since 𝜑 is multiplicative and nonzero, it is clear that 𝜑(1)=1. Now we are going to see that, for every 𝑓∈𝐶¹_𝑏(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≥𝜀 for some 𝜀>0. Thus 1/(𝑓−𝛼)²∈𝐶¹_𝑏(𝑀) and 1=𝜑𝑓−𝛼²⋅1𝑓−𝛼²=𝜑𝑓−𝛼²1⋅𝜑𝑓−𝛼². (53) 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)|≤sup_𝑥∈𝑀|𝑓(𝑥)| for every 𝑓∈𝐶¹_𝑏(𝑀), so we deduce that ‖𝜑‖=1.
We endow 𝔐(𝑀) with the weak* topology it inherits from the dual space 𝐶¹_𝑏(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)=𝛿_𝑥, where 𝛿_𝑥(𝑓)=𝑓(𝑥) for every 𝑓∈𝐶¹_𝑏(𝑀) and 𝑥∈𝑀. Note that every 𝐶¹-smooth function 𝑓∶𝑀→ℝ with compact support belongs to 𝐶¹_𝑏(𝑀), and thus, in particular, 𝐶¹_𝑏(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 (𝑥_𝜄) in 𝑀 converges to 𝑥 if and only if the net of evaluations (𝛿_𝑥𝜄) converges to 𝛿_𝑥 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 𝑊=𝜓∈𝔐(𝑀)∶𝜓𝑓_𝑗−𝜑𝑓_𝑗<𝜀,forj=1,…,𝑚, (54) where 𝑓₁,…,𝑓_𝑚∈𝐶¹_𝑏(𝑀) and 𝜀>0. Then there is some 𝑥∈𝑀 such that 𝛿_𝑥∈𝑊, since otherwise the function 𝑔=∑^𝑚_𝑗=1(𝑓_𝑗−𝜑(𝑓_𝑗))²∈𝐶¹_𝑏(𝑀) would satisfy 𝑔≥𝜀² 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 some 𝑥∈𝑀 such that 𝜑=𝛿_𝑥.

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 (𝑥_𝑛) in 𝑀 such that (𝛿_𝑥𝑛) converges to 𝜑. Since 𝜑∉𝛿(𝑀), we see that (𝑥_𝑛) has no convergent subsequence in 𝑀. Since 𝑔 is proper, we deduce that lim_𝑛→∞|𝑔(𝑥_𝑛)|=+∞. Then there is a subsequence (𝑥_𝑛𝑘) such that |𝑔(𝑥_𝑛𝑘+1)|>1+|𝑔(𝑥_𝑛𝑘)| for every 𝑘. Now we can choose a 𝐶¹ function 𝜃∶ℝ→[0,1], with bounded derivative, such that 𝜃(𝑔(𝑥_𝑛2𝑘+1))=1 and 𝜃(𝑔(𝑥_𝑛2𝑘))=0 for every 𝑘. Then the function 𝑓=𝜃∘𝑔 belongs to 𝐶¹_𝑏(𝑀), but the sequence (𝛿_𝑥𝑛𝑘(𝑓)) is not convergent, which is a contradiction.
(2)⇒(1) Conversely, if 𝜑=𝛿_𝑥 for some 𝑥∈𝑀, consider a countable neighborhood basis (𝑉_𝑛) of 𝑥 in 𝑀. Then the family of closures {cl_𝔐(𝑀)𝑉_𝑛} is easily seen to be a countable neighborhood basis of 𝛿_𝑥 in 𝔐(𝑀) as required.

The following Lemma shows the metric properties of the embedding 𝛿∶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 1𝐶min1,𝑑_𝐹𝑥,𝑦≤𝛿_𝑥−𝛿_𝑦≤𝑑_𝐹𝑥,𝑦. (55)

Proof. Recall that 𝛿_𝑥−𝛿_𝑦=sup𝑓(𝑥)−𝑓𝑦∶𝑓∈𝐶¹_𝑏(𝑀);𝑓_𝐶1𝑏≤1. (56) Thus by the mean value inequality contained in Theorem 5 we deduce at once that ‖𝛿_𝑥−𝛿_𝑦‖≤𝑑_𝐹(𝑥,𝑦). For the other inequality, suppose that 𝑥≠𝑦, and consider the function Φ∶𝑀→ℝ defined by Φ(𝑢)=1𝐶min1,𝑑_𝐹(𝑥,𝑢). (57) It is clear that 0≤Φ≤1/𝐶≤1, and from Theorem 11 we have that Lip(Φ)≤1. Now given 0<𝜀<(1/2𝐶)min{1,𝑑_𝐹(𝑥,𝑦)}, by Theorem 8, there exists a 𝐶¹-smooth function 𝑓∶𝑀→ℝ such that |𝑓(𝑢)−Φ(𝑢)|≤𝜀, for every 𝑢∈𝑀, and Lip(𝑓)≤1+𝜀. Thus ‖𝑓‖_𝐶1𝑏≤1+𝜀. Now we consider 𝑓=(1/(1+𝜀))𝑓, and we have that ‖𝑓‖_𝐶1𝑏≤1. Furthermore, 𝑓(𝑥)−𝑓𝑦=1𝑓(𝑥)𝑦≥11+𝜀−𝑓Φ(𝑥)𝑦=11+𝜀−Φ−2𝜀11+𝜀𝐶min1,𝑑_𝐹𝑥,𝑦−2𝜀,1+𝜀 (58) and the result follows.

Recall that, if 𝑀 and 𝑁 are Finsler manifolds, a mapping 𝑇∶𝐶¹_𝑏(𝑁)→𝐶¹_𝑏(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∈𝐶¹_𝑏(𝑁). Now we give the main result in this section, which provides a characterization of such normed algebra isomorphisms.

Theorem 16. Let (𝑀,𝐹_𝑀) and (𝑁,𝐹_𝑁) be second countable, connected, (forward) complete, quasi-reversible Finsler manifolds, with constants 𝐶_𝑀 and 𝐶_𝑁, respectively. For a mapping 𝑇∶𝐶¹_𝑏(𝑁)→𝐶¹_𝑏(𝑀), the following are equivalent: (1)𝑇 is a normed algebra isomorphism.(2)There exists a 𝐶¹-diffeomorphism, ℎ∶𝑀→𝑁, which is bi-Lipschitz for the respective Finsler distances, such that 𝑇(𝑓)=𝑓∘ℎ, for every 𝑓∈𝐶¹_𝑏(𝑁).
Moreover, in this case, the bi-Lipschitz constant of ℎ can be chosen to be max𝐶_𝑁⋅‖𝑇‖,𝐶_𝑀⋅𝑇⁻¹. (59)

Proof. (1)⇒(2) Suppose that 𝑇∶𝐶¹_𝑏(𝑁)→𝐶¹_𝑏(𝑀) is an isomorphism of normed algebras. Consider the transpose map 𝑇^∗∶𝐶¹_𝑏(𝑀)^∗→𝐶¹_𝑏(𝑁)^∗, defined by 𝑇^∗(𝜑)=𝜑∘𝑇 for every 𝜑∈𝐶¹_𝑏(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 𝛿_𝑀∶𝑀→𝔐(𝑀) and 𝛿_𝑁∶𝑁→𝔐(𝑁). By Lemma 14 we deduce that 𝑇^∗(𝛿_𝑀(𝑀))=𝛿_𝑁(𝑁), so that the restriction of 𝑇^∗ defines a homeomorphism from 𝛿_𝑀(𝑀) onto 𝛿_𝑁(𝑁). Thus we can define ℎ∶=(𝛿_𝑁)⁻¹∘𝑇^∗∘𝛿_𝑀∶𝑀→𝑁, which is a homeomorphism from 𝑀 onto 𝑁. Furthermore, we have that, for every 𝑥∈𝑀 and 𝑓∈𝐶¹_𝑏(𝑁), 𝑇𝑓(𝑥)=𝛿_𝑥𝑇𝑓=𝑇^∗𝛿_𝑥𝑓=𝛿_ℎ(𝑥)𝑓=𝑓(ℎ(𝑥)), (60) 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 that ℎ∶𝑀→𝑁 is bi-Lipschitz for the respective Finsler distances 𝑑_𝑀 and 𝑑_𝑁. Using Proposition 6, it will suffice to prove that ℎ and ℎ⁻¹ are locally Lipschitz. Given 𝑝∈𝑀, consider the open neighborhood 𝑈^𝑝=𝐁⁺_𝑝12∩𝐁⁻_𝑝12∩ℎ⁻¹𝐁⁺_ℎ𝑝12∩ℎ⁻¹𝐁⁻_ℎ𝑝12. (61) If 𝑥,𝑦∈𝑈^𝑝 we have that 𝑑_𝑀(𝑥,𝑦)<1 and 𝑑_𝑁(ℎ(𝑥),ℎ(𝑦))<1, so by Lemma 15 and taking into account that ‖𝑇^∗‖=‖𝑇‖, we obtain that 1𝐶_𝑁𝑑_𝑁ℎ(𝑥),ℎ𝑦≤𝛿_ℎ(𝑥)−𝛿_ℎ(𝑦)=𝑇^∗𝛿_𝑥−𝛿_𝑦≤‖𝑇‖⋅𝛿_𝑥−𝛿_𝑦≤‖𝑇‖𝑑_𝑀𝑥,𝑦. (62) Therefore, we have that ℎ is 𝐶_𝑁⋅‖𝑇‖-Lipschitz. In the same way, ℎ⁻¹ is 𝐶_𝑀⋅‖𝑇⁻¹‖-Lipschitz and the result follows.
(2)⇒(1) Taking into account Theorem 5, we have that 𝐶¹_𝑏(𝑀) is the set of all 𝐶¹-smooth functions 𝑓∶𝑀→ℝ which are Lipschitz, and the same holds for 𝐶¹_𝑏(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 mapping ℎ∶(𝑀,𝐹_𝑀)→(𝑁,𝐹_𝑁) 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∈𝑇_𝑥𝑀, 𝐹_𝑀(𝑥,𝑣)=𝐹_𝑁(ℎ(𝑥),𝑑ℎ(𝑥)(𝑣)). (63)

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,𝐹_𝑀) and (𝑁,𝐹_𝑁) 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 𝑑_𝑀 and 𝑑_𝑁, that is, for every 𝑥,𝑦∈𝑀, 𝑑_𝑀𝑥,𝑦=𝑑_𝑁ℎ(𝑥),ℎ𝑦. (64)

Now combining the above result with Theorem 16 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. Let 𝑀 and 𝑁 be second countable, connected, reversible, and complete Finsler manifolds. Then 𝑀 and 𝑁 are equivalent as Finsler manifolds if and only if 𝐶¹_𝑏(𝑀) and 𝐶¹_𝑏(𝑁) are isometric normed algebras. Moreover, every normed algebra isometry 𝑇∶𝐶¹_𝑏(𝑁)→𝐶¹_𝑏(𝑀) is of the form 𝑇(𝑓)=𝑓∘ℎ, where ℎ∶𝑀→𝑁 is a Finsler isometry.

Acknowledgments

M. I. Garrido and J. A. Jaramillo have been supported in part by D.G.I. (Spain) Grant MTM2009-07848. Y. C. Rangel has been associated to the Project 014-CT-2012 (CDCHT-UCLA) (Venezuela).