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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 164571, 10 pages
http://dx.doi.org/10.1155/2013/164571
Research Article

Smooth Approximation of Lipschitz Functions on Finsler Manifolds

1Instituto de Matemática Interdisciplinar (IMI), Departamento de Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain
2Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
3Departamento de Matemática, Universidad Centroccidental Lisandro Alvarado, Barquisimeto 3001, Estado de Lara, Venezuela

Received 13 May 2013; Accepted 24 June 2013

Academic Editor: Miguel Martin

Copyright © 2013 M. I. Garrido et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 defined on a connected, second countable Finsler manifold , for each positive continuous function and each , there exists a -smooth Lipschitz function such that , for every , and . As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebra of all functions with bounded derivative on a complete quasi-reversible Finsler manifold , we obtain a characterization of algebra isomorphisms 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 -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 -fine topology, by -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 -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 of all functions with bounded derivative on a quasi-reversible Finsler manifold , and we obtain a characterization of normed algebra isomorphisms 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 be a finite-dimensional real vector space. One says that a functional is a Minkowski norm on if the following conditions are satisfied (i) Positivity: if and only if .(ii) Triangle inequality: , for every .(iii)Positive homogeneity: , for every and every .(iv) Regularity: is continuous on and -smooth on .(v) Strong convexity: for every , the quadratic form associated to the second derivative of the function at , that is, is positive definite on .

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 is symmetric or absolutely homogeneous if In this case, is a norm in the usual sense.

Now the definition of Finsler manifold is as follows.

Definition 2. A Finsler manifold is a pair , where is a finite-dimensional -smooth manifold and is a continuous function defined on the tangent bundle , satisfying (i) is -smooth on .(ii)For every , is a Minkowski norm on the tangent space .

In particular, a Riemannian manifold is a special case of Finsler manifold, where the Minkowski norm on each tangent space is given by an inner product. The Finsler structure is said to be reversible if, for every , is symmetric. This is of course the case of Riemannian manifolds.

Now suppose that is a connected Finsler manifold. The Finsler distance   on is defined by where the Finsler length of a piecewise path is defined as: In this way we have (see Section 6.2 of [6]) that the Finsler distance is the so-called an asymmetric distance on , in the sense that it verifies (i).(ii) if and only if .(iii), for every .

In general, needs not to be symmetric. Nevertheless, when is reversible the Finsler distance is symmetric, and therefore is a metric space in the usual sense. In general, for each and , the forward ball of center and radius is defined as In the same way, the backward ball of center and radius is defined as 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 .

If is a Minkowski space, that is, a vector space endowed with a Minkowski norm, then the associated asymmetric distance is given by In this case, we will denote by the forward ball of center and radius , and we call it the Minkowski ball of center and radius . That is,

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 be a connected Finsler manifold, let , and consider such that the exponential mapping is a -diffeomorphism. Then, for with , one has that

A terminological remark is now in order. Suppose that and are two nonempty sets, endowed with asymmetric distances and , respectively. We say that a mapping is Lipschitz, with constant (or, briefly, -Lipschitz) if, for every , As usual, we will say that is -bi-Lipschitz when is bijective and both and are -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 be a connected Finsler manifold. For each and each , there exists such that the exponential mapping is a -diffeomorphism, which is -bi-Lipschitz.

3. Mean Value Inequality

In this section we obtain a kind of mean value inequality in the context of Finsler manifolds. If is a connected Finsler manifold, we define the Lipschitz constant of a function as Of course is Lipschitz if and only if . We denote by the space of all real Lipschitz functions defined on . If is now a -smooth function, we define as usual the norm of its differential at a point by Next, we give the following result providing the desired mean value inequality.

Theorem 5. Let be a connected Finsler manifold and a -smooth function. Then, Thus, for every , one has that

Proof. For the proof, fix a number , and we are going to see that the following conditions are equivalent: (1) is -Lipschitz. (2), for each .
Suppose there exists some with . Then there is some such that and . Suppose, for example, that , the other case being analogous. As it is shown [6] (see Theorem  6.3.1), we can choose such that the geodesic , defined for , minimizes the Finsler distance from point , that is, , for every . Now define by . We then have that , and therefore there exists some such that Since , choosing , we obtain that which contradicts the fact that is -Lipschitz.
Consider , and let . Choose a piecewise path such that , , and Define now by . Then, This shows that , for every .

We finish this section with the following simple result giving a local characterization of Lipschitz mappings, which will be useful later.

Proposition 6. Let and be connected Finsler manifolds, with Finsler distances and , respectively. A mapping is -Lipschitz if and only if it is locally -Lipschitz; that is, every point has a neighborhood such that, for every ,

Proof. Suppose that is locally -Lipschitz. Consider and . Choose a piecewise path from to , such that . Each has an open neighborhood where is -Lipschitz. Choose a partition of such that, for every ,   is contained into for some . Then In this way we obtain that is -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 is a Lipschitz function defined on a connected and second-countable Finsler manifold and is given, there exists a -smooth Lipschitz function such that , for every , and . We start with the following simple Lemma, which gives a first result of smooth approximation in Minkowski spaces.

Lemma 7. Let be a vector space endowed with a Minkowski norm. Consider an open set and, for , denote Suppose that is Lipschitz and let . Then there exists a -smooth function such that , for every , and .

Proof. By choosing a basis of , we may assume that . Note that, by local compactness, it follows that the Minkowski norm is equivalent to the usual Euclidean norm in , in the sense that there exists some such that for every . Now, if is a -Lipschitz function for the Minkowski norm , then is a -Lipschitz function for the Euclidean norm. Hence, using, for example, the well-known MacShane extension result, we can obtain a Lipschitz extension .
Now consider a sequence of usual -smooth mollifiers on , where each is nonnegative, is contained in the Euclidean ball , and . For each , define by Each is -smooth, and, since is uniformly continuous, we have that the sequence converges to uniformly on . Given , choose and large enough so that and define . Then, if , Therefore, we have that , as we wanted.

We next give the main result of the paper.

Theorem 8. Let be a connected and second countable Finsler manifold, let be a Lipschitz function, consider a continuous function, and let . Then there is a -smooth Lipschitz function such that , for every , and .

Proof. Let us denote . Without loss of generality we may assume that, for every , is small enough so that and Using Corollary 4, for each , we can choose such that the exponential mapping is a -diffeomorphism and -bi-Lipschitz from the Minkowski ball onto the forward ball , where denotes the null vector of . In addition, by the continuity of and , we can also assume that and , for every . Since is second countable, there is a sequence in such that where we denote . Now, for each , we define by and we then have that is -Lipschitz.
Next, we are going to construct a partition of unity subordinated to the covering of , estimating the Lipschitz constant of the respective functions. Thus, for each , let be a -smooth function such that and define by It is clear that each is -smooth and Lipschitz. Furthermore, on the forward ball and on . Now we define the functions by setting and, for , Then, it is easy to check that, for every , (i) is -smooth and -Lipschitz, where . (ii). (iii) on , whenever .
Thus, is the desired partition of unity. Indeed, for each , let be the first integer such that . Then and , for . Therefore, the family is locally finite. In addition , since Now using Lemma 7 we can find, for each , a -smooth function such that for every , and Thus, we define the approximation function by for each . Note that, since the exponential mapping is a -diffeomorphism from onto , the expression is well defined for and it is -smooth on . On the other hand, if , then . Thus, for every , we may assume that is zero. With this convention, and taking into account that is a -smooth partition of unity, we obtain that is well defined and -smooth on .
We are going to see that is also Lipschitz and that and approximate to and , respectively. Fix , and consider again the first integer such that . To simplify, denote , for all . Then we have Finally, let us check that is -Lipschitz, and hence . By Proposition 6, it will suffice to see that is locally -Lipschitz. Fix and, as before, consider the first integer such that . We are going to see that is -Lipschitz on the open set , where For each , denote Then, whenever , the following holds:(1) if and , then . Indeed, if , then . Thus if we have that and therefore .(2) For every we have that . That is clear from the above. In particular, if then and are well defined, and, using (33) and the fact that is -Lipschitz, we see that (3) If then . That follows, since and , for every .
As a consequence of the above we have, for every , and using the notation and , the following: (i), (ii), (iii), (iv), whenever .
Therefore, since we deduce that Finally, using (25), (32), and the fact that is -Lipschitz, we then have that since . This shows that is locally -Lipschitz and we finish the proof.

Remark 9. In general, the exponential mapping in a Finsler manifold is only -smooth. According to a result of Akbar-Zadeh in [8] (see also [6], page 127), the exponential mapping is -smooth if and only if it is -smooth, and this property characterizes a special class of Finsler manifolds, called manifolds of Berwald type. Thus, if is a connected, second countable manifold of Berwald type, the same proof above gives that the approximating function in Theorem 8 can be chosen to be -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 is said to be quasi-reversible if there exists some such that

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 . 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 is a manifold of Berwald type, then all its tangent spaces , for every , are linearly isometric to each other.

We next give a useful characterization of connected quasi-reversible manifolds.

Theorem 11. Let be a connected Finsler manifold, and let . The following conditions are equivalent: (1), for every .(2), for every .(3)For all , the forward distance function is -Lipschitz.(4)For all , the backward distance function is -Lipschitz.

Proof. Let and consider a piecewise path from to . Then the reverse path given by is a piecewise path from to . Now, This implies that . Interchanging the roles of and , we obtain the reverse inequality.
Let . From Theorem 3 we have that if then That is, Thus we obtain that For every we have, from the triangle inequality, that By the hypothesis (2), we follow at once that This means that the function is -Lipschitz.
For every we have that, in particular, Choosing we have that Reversing the roles of and , we also have that .
This can be seen as before.

Note that, by choosing in the above result, we can deduce at once the following characterization of connected reversible manifolds.

Corollary 12. Let be a connected Finsler manifold. The following conditions are equivalent: (1), for every , (2), for every .

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 in a Finsler manifold is said to be forward Cauchy (respectively, backward Cauchy) if, for every , there exists some such that, if , then , (respectively, ). We say then that 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 is said to be proper if, for every compact set , its preimage is compact.

Theorem 13. Let be a connected, second countable, and quasi-reversible Finsler manifold. The following conditions are equivalent: (1) is forward complete. (2)There exists a proper Lipschitz function . (3)There exists a proper -smooth function whose differential is uniformly bounded in norm.

Proof. Fix and consider the forward distance function . The Hopf-Rinow Theorem (see Theorem  6.6.1 in [6]) gives that is forward complete if and only if every closed and (forward) bounded subset of is compact. This implies that is a proper function. Furthermore, by Theorem 11, we have that is Lipschitz.
Suppose that the proper function is -Lipschitz, and fix some . By Theorem 8, there exists a -smooth function such that (i), for every . (ii).
It is easy to check that is a proper function, since is so. On the other hand, by Theorem 5, we have that , for every .
Let be a forward Cauchy sequence in . Since is Lipschitz, then is a Cauchy sequence in , and therefore converges to some point in . Thus is a compact subset of . Now is contained in , which is compact since is proper. Then is convergent in .

6. Algebras of Differentiable Functions on Finsler Manifolds

The classical Myers-Nakai Theorem asserts that the Riemannian structure of a Riemannian manifold is determined by the natural normed algebra structure on the space of all bounded functions on which have bounded derivative (or, equivalently, which are Lipschitz on with respect to the geodesic distance). This was proved by Myers [10] in the case that 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 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 be a Finsler manifold, and let denote the space of all real bounded -smooth functions defined on whose derivative has uniformly bounded norm. We endow with the natural norm: Endowed with this norm, is a complete normed algebra. Note that is not submultiplicative, but it satisfies that .

Next we are going to recall the definition of the structure space associated to . This construction is standard, but we give some details for the reader’s convenience. Let denote the set of all nonzero, multiplicative, continuous linear forms .

Claim. Each satisfies that , and furthermore is positive, that is, whenever .

Proof. Since is multiplicative and nonzero, it is clear that . Now we are going to see that, for every , we have that belongs to the closure of . Indeed, if is not in the closure of , then for some . Thus and But we have that , which is a contradiction. From this we obtain that is positive. We also obtain that for every , so we deduce that .
We endow with the weak* topology it inherits from the dual space . Since is a weak*-closed subset of the unit ball, we see that is a compact space. Now we consider the embedding given by , where for every and . Note that every -smooth function with compact support belongs to , and thus, in particular, separates points and closed sets of . From this it is not difficult to deduce that is a topological embedding, that is, a net in converges to if and only if the net of evaluations converges to in the weak* topology. On the other hand, the set of evaluations is dense in . Indeed, let , and consider a weak* basic neighborhood of of the form where and . Then there is some such that , since otherwise the function would satisfy and , and this is impossible since is positive. In this way we see that is a compactification of .

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 be a connected, second countable, (forward) complete, quasi-reversible Finsler manifold, and let . The following conditions are equivalent: (1) has a countable neighborhood basis in . (2)There exists some such that .

Proof. Since is (forward) complete, by Theorem 13, there exists a proper -smooth function whose differential is uniformly bounded in norm. Suppose now that has a countable neighborhood basis in . Since is dense in , there is a sequence in such that converges to . Since , we see that has no convergent subsequence in . Since is proper, we deduce that . Then there is a subsequence such that for every . Now we can choose a function , with bounded derivative, such that and for every . Then the function belongs to , but the sequence is not convergent, which is a contradiction.
Conversely, if for some , consider a countable neighborhood basis of in . Then the family of closures is easily seen to be a countable neighborhood basis of in as required.

The following Lemma shows the metric properties of the embedding .

Lemma 15. Let be a connected, second countable (forward) complete, quasi-reversible Finsler manifold, with constant . Then, for each , we have that

Proof. Recall that Thus by the mean value inequality contained in Theorem 5 we deduce at once that . For the other inequality, suppose that , and consider the function defined by It is clear that , and from Theorem 11 we have that . Now given , by Theorem 8, there exists a -smooth function such that , for every , and . Thus . Now we consider , and we have that . Furthermore, and the result follows.

Recall that, if and are Finsler manifolds, a mapping is said to be a normed algebra isomorphism provided is a bicontinuous linear bijection such that for every . Now we give the main result in this section, which provides a characterization of such normed algebra isomorphisms.

Theorem 16. Let and be second countable, connected, (forward) complete, quasi-reversible Finsler manifolds, with constants and , respectively. For a mapping , the following are equivalent: (1) is a normed algebra isomorphism.(2)There exists a -diffeomorphism, , which is bi-Lipschitz for the respective Finsler distances, such that , for every .
Moreover, in this case, the bi-Lipschitz constant of can be chosen to be

Proof. Suppose that is an isomorphism of normed algebras. Consider the transpose map , defined by for every . Since is multiplicative and is weak*-to-weak* bicontinuous, we see that the restriction of defines a homeomorphism from onto . Consider now the natural embeddings and . By Lemma 14 we deduce that , so that the restriction of defines a homeomorphism from onto . Thus we can define , which is a homeomorphism from onto . Furthermore, we have that, for every and , that is, . In particular, note that is -smooth for every -smooth function with compact support. From this it is easily deduced that is -smooth, and the same can be said about , so that is a -diffeomorphism.
Now we are going to see that is bi-Lipschitz for the respective Finsler distances and . Using Proposition 6, it will suffice to prove that and are locally Lipschitz. Given , consider the open neighborhood If we have that and , so by Lemma 15 and taking into account that , we obtain that Therefore, we have that is -Lipschitz. In the same way, is -Lipschitz and the result follows.
Taking into account Theorem 5, we have that is the set of all -smooth functions which are Lipschitz, and the same holds for , 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 between Finsler manifolds is said to be a Finsler isometry if is a diffeomorphism which preserves the Finsler structure, that is, for every and every ,

Extending the classical result by Myers and Steenrod [14] about Riemannian manifolds, Deng and Hou have proved in ([7], Theorem 2.2) that, if and are connected Finsler manifolds, a mapping is a Finsler isometry if and only if is bijective and preserves the corresponding Finsler distances and , that is, for every ,

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 and be second countable, connected, reversible, and complete Finsler manifolds. Then and are equivalent as Finsler manifolds if and only if and are isometric normed algebras. Moreover, every normed algebra isometry is of the form , where 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).

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