Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions

In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^{k}$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete $C^{k}$ Finsler manifold $M$ is determined by the algebra $C_b^k(M)$; (ii) the weak Finsler structure of a separable and complete $C^{k}$ Finsler manifold $M$ modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function is determined by the algebra $C^k_b(M)$; (iii) the weak Finsler structure of a $C^k$ uniformly bumpable and complete $C^{k}$ Finsler manifold $M$ modeled on a Weakly Compactly Generated (WCG) Banach space with an (equivalent) $C^k$ smooth norm is determined by the algebra $C^k_b(M)$; and (iv) the isometric structure of a WCG Banach space $X$ with an $C^1$ smooth bump function is determined by the algebra $C_b^1(X)$.

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Introduction and Preliminaries
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In this note, we are interested in characterizing the Finsler structure of a Finsler manifold $M$ in terms of the space of real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. The problem of the interrelation of the topological, metric and smooth structure of a space $X$ and the algebraic and topological structure of the space $C(X)$ (the set of real-valued continuous functions defined on $X$) has been largely studied. These results are usually referred as *Banach-Stone type theorems*. Recall the celebrated Banach-Stone theorem, asserting that the compact spaces $K$ and $L$ are homeomorphic if and only if the Banach spaces $C(K)$ and $C(L)$ endowed with the sup-norm are isometric. For more information on Banach-Stone type theorems see the survey @GaJaExtracta and references therein.

The Myers-Nakai theorem states that the structure of a complete Riemannian manifold $M$ is characterized in terms of the Banach algebra $C^1_b(M)$ of all real-valued, bounded and $C^1$ smooth functions with bounded derivative defined on $M$ endowed with the sup-norm of the function and its derivative. More specifically, two complete Riemannian manifolds $M$ and $N$ are equivalent as Riemannian manifolds, i.e. there is a $C^1$ diffeomorphism $h:M\to N$ such that $$\langle dh(x)(v),dh(x)(w)\rangle_{h(x)}=\langle v,w \rangle_x$$ for every $x\in M$ and $v,w\in T_xM$ if and only if the Banach algebras $C^1_b(M)$ and $C^1_b(N)$ are isometric. This result was first proved by S. B. Myers @Myers for a compact and Riemannian manifold and by M. Nakai @Nakai for a finite-dimensional Riemannian manifold. Very recently, I. Garrido, J.A. Jaramillo and Y.C. Rangel @GaJaRa gave an extension of the Myers-Nakai theorem for every infinite-dimensional, complete Riemannian manifold. A similar result for the so-called finite-dimensional Riemannian-Finsler manifolds is given in @GaJaRa2 (see also @Rangel-Tesis).

Our aim in this work is to extend the Myers-Nakai theorem to the context of Finsler manifolds. On the one hand, we obtain the Myers-Nakai theorem for (i) finite-dimensional and complete Finsler manifolds, and (ii) WCG Banach spaces with a $C^1$ smooth bump function. On the other hand, we study for $k\ge 1$ the algebra $C^k_b(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded first derivative defined on a complete Finsler manifold $M$. We prove that these algebras determine the weak Finsler structure of a complete Finsler manifold when $k=1$ and the Finsler structure when $k\ge 2$. In particular, we obtain a weaker version of the Myers-Nakai theorem for (i) separable and complete Finsler manifolds modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function, and (ii) $C^k$ uniformly bumpable and complete Finsler manifolds modeled on WCG Banach spaces with an equivalent $C^k$ smooth norm. In the proof of these results we will use the ideas of the Riemannian case @GaJaRa.

The notation we use is standard. The norm in a Banach space $X$ is denoted by $||\cdot||$. The dual space of $X$ is denoted by $X^*$ and its dual norm by $||\cdot||^*$. The open ball with center $x\in X$ and radius $r>0$ is denoted by $B(x,r)$. A $C^k$ smooth bump function $b:X\to {\mathbb{R}}$ is a $C^k$ smooth function on $X$ with bounded, non-empty support, where ${\operatorname{supp}}(b) =\overline{\{x \in  X : b(x)\neq 0\}}$. If $M$ is a Banach manifold, we denote by $T_xM$ the tangent space of $M$ at $x$. Recall that the tangent bundle of $M$ is $TM=\{(x,v):x\in M \text{ and } v\in T_x M\}$. We refer to @DGZ, @fabianhajek, @Lang and @Deim for additional definitions. We will say that the norms $||\cdot||_{1}$ and $||\cdot||_{2}$ defined on a Banach space $X$ are $K$-equivalent ($K\ge 1$) whether $\frac{1}{K}||v||_1\le||v||_2\le K||v||_1$, for every $v\in X$.

Let us begin by recalling the definition of a $C^k$ Finsler manifold in the sense of Palais as well as some basic properties (for more information about these manifolds see @Palais, @Deim, @Rabier, @Neeb, @GaGuJa and @MarLuis).

\[defFinsler\] Let $M$ be a (paracompact) $C^k$ Banach manifold modeled on a Banach space $(X,||\cdot||)$, where $k \in{\mathbb{N}}\cup \{\infty\}$. Let us consider the tangent bundle $TM$ of $M$ and a continuous map $||\cdot||_M: TM\to [0,\infty)$. We say that $(M,||\cdot||_M)$ is a **$C^k$ Finsler manifold in the sense of Palais** if $||\cdot||_M$ satisfies the following conditions:

1.  For every $x\in M$, the map $||\cdot||_x:={||\cdot||_M}_{\mid_{T_xM}}:T_xM\to [0,\infty)$ is a norm on the tangent space $T_xM$ such that for every chart $\varphi:U\to X$ with $x \in U$, the norm $v\in X \mapsto ||d\varphi^{-1}(\varphi(x))(v)||_x$ is equivalent to $||\cdot||$ on $X$.

2.  For every $x_0\in M$, every $\varepsilon>0$ and every chart $\varphi:U\to X$ with $x_0\in U$, there is an open neighborhood $W$ of $x_0$ such that if $x\in W$ and $v\in X$, then $$\label{palaisdef}
    \frac{1}{1+\varepsilon}||d\varphi^{-1}(\varphi(x_0))(v)||_{x_0}\le ||d\varphi^{-1}(\varphi(x))(v)||_{x}\le (1+\varepsilon)||d\varphi^{-1}(\varphi(x_0))(v)||_{x_0}.$$ In terms of equivalence of norms, the above inequalities yield the fact that the norms $||d\varphi^{-1}(\varphi(x))(\cdot)||_{x}$ and $||d\varphi^{-1}(\varphi(x_0))(\cdot)||_{x_0}$ are $(1+\varepsilon)$-equivalent.

Let us recall that Banach spaces and Riemannian manifolds are $C^\infty$ Finsler manifolds in the sense of Palais @Palais.

Let $M$ be a Finsler manifold, we denote by $T_xM^*$ the dual space of the tangent space $T_xM$. Let $f:M\to{\mathbb{R}}$ be a differentiable function at $p\in M$. The norm of $df(p)\in {T_p M}^*$ is given by $$||df(p)||_p= \sup\{|df(p)(v)|:v\in T_p M, ||v||_p\le 1\}.$$ Let us consider a differentiable function $f:M\to N$ between Finsler manifolds $M$ and $N$. The norm of the derivative at the point $p\in M$ is defined as $$\begin{aligned}
& ||df(p)||_p= \sup\{||df(p)(v)||_{f(p)}:v\in T_p M, ||v||_p\le 1\} = &\\
& \qquad \qquad =\sup\{\xi(df(p)(v)):\xi\in {T_{f(p)} N}^*,\ v\in T_pM\ \text{and}\ ||v||_{p}=1=||\xi||^*_{f(p)} \},&\end{aligned}$$ where $||\cdot||_{f(p)}^*$ is the dual norm of $||\cdot||_{f(p)}$. Recall that if $(M, ||\cdot||_M)$ is a Finsler manifold, the <span>*length*</span> of a piecewise $C^1$ smooth path $c:[a,b]\rightarrow M$ is defined as $\ell(c):=\int_{a}^b||c'(t)||_{c(t)}\,dt$. Besides, if $M$ is connected, then it is connected by piecewise $C^1$ smooth paths, and the associated <span>*Finsler metric*</span> $d_M$ on $M$ is defined as $$d_M(p,q)=\inf\{\ell(c): \, c \text{ is a piecewise } C^1 \text{ smooth path connecting } p \text{ and } q\}.$$ It was shown in @Palais that the Finsler metric is consistent with the topology given in $M$. The open ball of center $p\in M$ and radius $r>0$ is denoted by $B_M(p,r):=\{q\in M:\, d_M(p,q)<r\}$. The Lipschitz constant ${\operatorname{Lip}}(f)$ of a Lipschitz function $f:M\rightarrow N$, where $M$ and $N$ are Finsler manifolds, is defined as ${\operatorname{Lip}}(f)=\sup\{\frac{d_N(f(x),f(y))}{d_M(x,y)}: x,y \in M, x\not=y\}$. We shall only consider connected manifolds. Let us recall the following "mean value inequality" for Finsler manifolds @AzFeMe [@MarLuis].

\[meanvalue\] Let $M$ and $N$ be $C^1$ Finsler manifolds (in the sense of Palais) and $f:M\to N$ a $C^1$ smooth function. Then, $f$ is Lipschitz if and only if $||df||_\infty := \sup\{||df(x)||_x:x\in M\}<\infty$. Furthermore, ${\operatorname{Lip}}(f)= ||df||_\infty$.

We will also need the following result related to the $(1+\varepsilon)$-bi-Lipschitz local behavior of the charts of a $C^1$ Finsler manifold in the sense of Palais @MarLuis [Lemma 2.4].

\[desigualdades:BiLipschitz\] Let us consider a $C^1$ Finsler manifold $M$ (in the sense of Palais). Then, for every $x_0\in M$ and every chart $(U,\varphi)$ with $x_0\in U$ satisfying inequality , there exists an open neighborhood $V\subset U$ of $x_0$ satisfying $$\label{B-L1}
\frac{1}{1+\varepsilon}d_M(p,q)\le |||\varphi(p)-\varphi(q)|||\le (1+\varepsilon) d_M(p,q), \quad \text{ for every } p,q\in V,$$ where $|||\cdot|||$ is the (equivalent) norm $||d\varphi^{-1}(\varphi(x_0))(\cdot)||_{x_0}$ defined on $X$.

Now, let us recall the concept of <span>*uniformly bumpable manifold*</span>, introduced by D. Azagra, J. Ferrera and F. López-Mesas @AzFeMe for Riemannian manifolds. A natural extension to <span>Finsler manifolds</span> is defined in the same way @MarLuis.

\[bumpable\] A $C^k$ Finsler manifold in the sense of Palais $M$ is **$C^k$ uniformly bumpable** whenever there are $R>1$ and $r>0$ such that for every $p\in M$ and $\delta\in (0,r)$ there exists a $C^k$ smooth function $b:M\to[0,1]$ such that:

1.  $b(p)=1$,

2.  $b(q)=0$ whenever $d_M(p,q)\ge \delta$,

3.  $\sup_{q\in M} ||d b(q)||_q \le R/\delta$.

Note that this is not a restrictive definition: D. Azagra, J. Ferrera, F. López-Mesas and Y. Rangel @AzFeMeRa proved that every separable Riemannian manifold is $C^\infty$ uniformly bumpable. This result was generalized in @MarLuis, where it was proved that every $C^1$ Finsler manifold (in the sense of Palais) modeled on a certain class of Banach spaces (such as Hilbert spaces, Banach spaces with separable dual, closed subspaces of $c_0(\Gamma)$ for every set $\Gamma\neq \emptyset$) is $C^1$ uniformly bumpable. In particular, every Riemannian manifold (either separable or non-separable) is $C^1$ uniformly bumpable.

It is straightforward to verify that if a $C^{k}$ Finsler manifold $M$ is modeled on a Banach space $X$ and $M$ is $C^k$ uniformly bumpable, then $X$ admits a Lipschitz $C^k$ smooth bump function. Besides, a <span>*separable*</span> $C^k$ Finsler manifold $M$ is modeled on a Banach space with a Lipschitz, $C^k$ smooth bump function if and only if $M$ is $C^k$ uniformly bumpable @MarLuis. Nevertheless, we do not know whether this equivalence holds in the non-separable case.

From now on, we shall refer to $C^{k}$ Finsler manifolds in the sense of Palais as $C^{k}$ Finsler manifolds, and $k \in {\mathbb{N}}\cup \{\infty\}$. We shall use the standard notation of $C^k(U,Y)$ for the set of all $k$-times continuously differentiable functions defined on an open subset $U$ of a Banach space (Finsler manifold) taking values into a Banach space (Finsler manifold) $Y$. We shall write $C^k(U)$ whenever $Y=\mathbb R$.

Now, let us recall the concept of weakly $C^k$ smooth function.

Let $X$ and $Y$ be Banach spaces and consider a function $f:U\to Y$, where $U$ is an open subset of $X$. The function $f$ is said to be **weakly $C^k$ smooth** at the point $x_0$ whenever there is an open neighborhood $U_{x_0}$ of $x_0$ such that $y^*\circ  f$ is $C^k$ smooth at $U_{x_0}$, for every $y^*\in Y^*$. The function $f$ is said to be **weakly $C^k$ smooth** on $U$ whenever $f$ is weakly $C^k$ smooth at every point $x\in U$.

On the one hand, J. M. Gutiérrez and J.L. G. Llavona @GULLA proved that if $f:U\to Y$ is weakly $C^k$ smooth on $U$, then $g\circ f\in C^k(U)$ for all $g\in C^k(Y)$. They also proved that if $f:U\to Y$ is weakly $C^k$ smooth on $U$, then $f\in C^{k-1}(U)$. For $k=1$, the above yields that every weakly $C^1$ smooth function on $U$ is continuous on $U$. Also, for $k=\infty$, every weakly $C^\infty$ smooth function on $U$ is $C^\infty$ smooth on $U$. M. Bachir and G. Lancien @BaLan proved that, if the Banach space $Y$ has the Schur property, then the concept of weakly $C^k$ smoothness coincides with the concept of $C^k$ smoothness. On the other hand, there are examples of weakly $C^1$ smooth functions that are not $C^1$ smooth (see @GULLA and @BaLan).

Let $M$ and $N$ be $C^{k}$ Finsler manifolds and $U\subset M$, $O\subset N$ open subsets of $M$ and $N$, respectively. A function $f:U\to N$ is said to be **weakly $C^k$ smooth** at the point $x_0$ of $U$ if there exist charts $(W,\varphi)$ of $M$ at $x_0$ and $(V,\psi)$ of $N$ at $f(x_0)$ such that $ \psi\circ f \circ \varphi^{-1}$ is weakly $C^k$ smooth at $\varphi(W)$. We say that $f:U\to N$ is **weakly $C^k$ smooth** in $U$ if $f$ is weakly $C^k$ smooth at every point $x\in U$. We say that a bijection $f:U\rightarrow O$ is a weakly $C^k$ diffeomorphism if $f$ and $f^{-1}$ are weakly $C^k$ smooth on $U$ and $O$, respectively. Notice that these definitions do not depend on the chosen charts.

Let us note that there are homeomorphisms which are weakly $C^1$ smooth but not differentiable. Indeed, we follow @GULLA [Example 3.9] and define $g:{\mathbb{R}}\to c_0({\mathbb{N}})$ and $h:c_0({\mathbb{N}})\to c_0({\mathbb{N}})$ by $g(t)=(0,\frac{1}{2}\sin(2t),\dots,\frac{1}{n}\sin(nt),\dots)$ and $h(x)=x+g(x_1)$ for every $t\in {\mathbb{R}}$ and $x=(x_1,\dots, x_n, \dots)\in c_0$. The function $h$ is an homeomorphism, $h^{-1}(y)=y-g(y_1)$ for every $y\in c_0$, and $h$ is weakly $C^1$ smooth on $c_0({\mathbb{N}})$. Notice that if $h$ were differentiable at a point $x\in c_0$ with $x_1=0$, then $$h'(x)(1,0,0, \dots)=(1,1,1,\dots)Ê\in \ell_\infty \setminus c_0,$$ which is a contradiction.

Now, let us consider different definitions of isometries between $C^k$ Finsler manifolds.

Let $(M,||\cdot||_M)$ and $(N,||\cdot||_N)$ be $C^{k}$ Finsler manifolds and a bijection $h:M\to N$.

-   We say that $h$ is a **metric isometry** for the Finsler metrics, if $$\label{metric-isometry}
    d_N(h(x),h(y))=d_M(x,y), \qquad \text{ for every } x,y \in M.$$

-   We say that $h$ is a **$C^{k}$ Finsler isometry** if it is a $C^{k}$ diffeomorphism satisfying $$||dh(x)(v)||_{h(x)}=||(h(x),dh(x)(v))||_N=||(x,v)||_M=||v||_x,$$ for every $x\in M$ and $v\in T_x M$. We say that the Finsler manifolds $M$ and $N$ are **$C^{k}$ equivalent as Finsler manifolds** if there is a $C^{k}$ Finsler isometry between $M$ and $N$.

-   We say that $h$ is a **weak $C^{k}$ Finsler isometry** if it is a weakly $C^k$ diffeomorphism and a metric isometry for the Finsler metrics. We say that the Finsler manifolds $M$ and $N$ are **weakly $C^k$ equivalent as Finsler manifolds** if there is a weak $C^{k}$ Finsler isometry between $M$ and $N$.

\[Finsler:Isometry\] Let $M$ and $N$ be $C^{k}$ Finsler manifolds. Let us assume that there is a $C^k$ diffeomorphism and metric isometry (for the Finsler metrics) $h:M\to N$. Then $h$ is a $C^k$ Finsler isometry.

Let us fix $x\in M$ and $y=h(x)\in N$. For every $\varepsilon>0$, there are $r>0$ and charts $\varphi:B_M(x,r)\subset M \to X$ and $\psi: B_N(y,r) \subset N \to Y$ satisfying inequalities and . Since $h:M \to N$ is a metric isometry, $h$ is a bijection from $B_M(x,r)$ onto $B_N(y,r)$.

Let us consider the equivalent norms on $X$ and $Y$ defined as $
|||\cdot|||_x:=||d\varphi^{-1}(\varphi(x))(\cdot)||_x $ and $|||\cdot|||_y=||d\psi^{-1}(\psi(y))(\cdot)||_y,$ respectively.

Since $h$ is a metric isometry, we obtain from Lemma \[desigualdades:BiLipschitz\], for $p,q$ in an open neighborhood of $\varphi(x)$, $$\begin{aligned}
|||\psi\circ h\circ \varphi^{-1}(p)-\psi\circ h\circ  \varphi^{-1}(q)|||_y  \le  (1+\varepsilon)d_N(h\circ \varphi^{-1}(p),  h\circ \varphi^{-1}(q)) = &\\
 = (1+\varepsilon)d_M(\varphi^{-1}(p),\varphi^{-1}(q))\le (1+\varepsilon)^2|||p-q|||_x. &\end{aligned}$$ Thus, $\sup\{|||d(\psi\circ h\circ \varphi^{-1})(\varphi(x))(w)|||_y: |||w|||_x\le 1 \}\le (1+\varepsilon)^2$. Now, for every $v\in T_x M$ with $v\neq 0$, let us write $w=d\varphi(x)(v)\in X$. We have $$\begin{aligned}
||dh(x)(v)||_y & =||d\psi^{-1}(\psi(y))d\psi(y)dh(x)(v)||_y=|||d (\psi\circ h) (x)(v)|||_y = \\
& = ||| d (\psi\circ h) (x) d\varphi^{-1}(\varphi(x))(w)|||_y
  = ||| d(\psi\circ h \circ \varphi^{-1}) (\varphi(x))(w)|||_y \le \\
  & \le (1+\varepsilon)^2 |||w|||_x=(1+\varepsilon)^2 ||v||_x.\end{aligned}$$ Since this inequality holds for every $\varepsilon>0$ and the same argument works for $h^{-1}$, we conclude that $||dh(x)(v)||_y=||v||_x$ for all $v\in T_xM$. Thus, $h$ is a $C^k$ Finsler isometry.

Let us now turn our attention to the <span>*Banach algebra*</span> $C_b^1(M)$, the algebra of all real-valued, $C^1$ smooth and bounded functions with bounded derivative defined on a $C^1$ Finsler manifold $M$, i.e. $$C_b^1(M)=\{f:M\to{\mathbb{R}}: f\in C^1(M),\ ||f||_\infty<\infty \text{ and } ||df||_\infty<\infty\},$$ where $ ||f||_\infty:=\sup\{|f(x)|:\, x\in M\}$ and $||df||_\infty:=\sup\{||df(x)||_x:\, x\in M\}$. The usual norm considered on $C_b^1(M)$ is $||f||_{C_b^1}=\max\{||f||_\infty,||df||_\infty\}$ for every $f\in C_b^1(M)$ and $(C_b^1(M),||\cdot||_{C_b^1(M)})$ is a Banach space. Let us notice that, by Lemma \[meanvalue\], we have $||df||_\infty = {\operatorname{Lip}}(f)$. Recall that $(C_b^1(M),2||\cdot||_{C_b^1(M)})$ is a Banach algebra.

For $2\le k \le  \infty$ and a $C^{k}$ Finsler manifold $M$, let us consider the algebra $C_b^k(M)$ of all real-valued, $C^k$ smooth and bounded functions that have bounded first derivative, i.e. $$C_b^k(M)=\{f:M\to{\mathbb{R}}: f\in C^k(M),\ ||f||_\infty<\infty \text{ and } ||df||_\infty<\infty\}=C^k(M)\cap C_b^1(M).$$ with the norm $||\cdot||_{C_b^1}$. Thus, $C_b^k(M)$ is a subalgebra of $C_b^1(M)$. Nevertheless, it is not a Banach algebra. A function $\varphi:C_b^k(M)\to {\mathbb{R}}$ ($1\le k\le \infty$) is said to be an *algebra homomorphism* whether for all $f,g\in C_b^k(M)$ and $\lambda,\eta \in {\mathbb{R}}$,

1.  $\varphi(\lambda f + \eta g)=\lambda \varphi(f)+ \eta \varphi(g)$, and

2.  $\varphi(f\cdot g)=\varphi(f)\varphi(g)$.

Let us denote by $H(C_b^k(M))$ the set of all nonzero algebra homomorphisms, i.e. $$H(C_b^k(M))=\{\varphi: C_b^k(M)\to {\mathbb{R}}: \varphi \text{ is an algebra homomorphism and } \varphi(1)=1\}.$$

Let us list some of the basic properties of the algebra $C_b^k(M)$ and the algebra homomorphisms $H(C_b^k(M))$. They can be checked as in the Riemannian case (see @GaJa, @GaJaRa and @Isbell).

-   If $\varphi\in H(C_b^k(M))$, then $\varphi\not=0$ if and only if $\varphi(1)=1$.

-   If $\varphi\in H(C_b^k(M))$, then $\varphi$ is positive, i.e. $\varphi(f)\ge 0$ for every $f\ge 0$.

-   If the $C^{k}$ Finsler manifold $M$ is modeled on a Banach space that admits a Lipschitz and $C^k$ smooth bump function, then $C_b^k(M)$ is a <span>*unital algebra that separates points and closed sets*</span> of $M$. Let us briefly give the proof for completeness. Let us take $x\in M$, and $C\subset M$ a closed subset of $M$ with $x\not\in C$. Let us take $r>0$ small enough so that $C\cap  B_M(x,r) =\emptyset$ and a chart $\varphi:B_M(x,r)\to X$ satisfying inequality . Let us take $s>0$ small enough so that $\varphi(x)\in B(\varphi(x),s)\subset \varphi(B(x,r/2))\subset X$ and a Lipschitz and $C^k$ smooth bump function $b:X\to{\mathbb{R}}$ with $b(\varphi(x))=1$ and $b(z)=0$ for every $z\not\in B(\varphi(x),s)$. Let us define $h:M\to {\mathbb{R}}$ as $h(p)=b(\varphi(p))$ for every $p\in B_M(x,r)$ and $h(p)=0$ otherwise. Then $h\in C_b^k(M)$, $h
     (x)=1$ and $h(c)=0$ for every $c\in C$.

-   The space $H(C_b^k(M))$ is closed as a topological subspace of ${\mathbb{R}}^{C_b^k(M)}$ with the product topology. Moreover, since every function in $C_b^k(M)$ is bounded, it can be checked that $H(C_b^k(M))$ is compact in ${\mathbb{R}}^{C_b^k(M)}$.

-   If $C_b^k(M)$ separates points and closed subsets, then $M$ can be embedded as a topological subspace of $H(C_b^k(M))$ by identifying every $x\in M$ with the *point evaluation homomorphism* $\delta_x$ given by $\delta_x(f)=f(x)$ for every $f\in C_b^k(M)$. Also, it can be checked that the subset $\delta(M)=\{\delta_x:x\in M\}$ is dense in $H(C_b^k(M))$. Therefore, it follows that $H(C_b^k(M))$ is a compactification of $M$.

-   Every $f\in C_b^k(M)$ admits a continuous extension $\widehat{f}$ to $H(C_b^k(M))$, where $\widehat{f}(\varphi)=\varphi(f)$ for every $\varphi\in H(C_b^k(M))$. Notice that this extension $\widehat{f}$ coincides in $H(C_b^k(M))$ with the projection $\pi_f:{\mathbb{R}}^{C_b^k(M)}\to {\mathbb{R}}$, given by $\pi_f(\varphi)=\varphi(f)$, i.e. ${\pi_f}_{\mid_{H(C_b^k(M))}}=\widehat{f}$. In the following, we shall identify $M$ with $\delta(M)$ in $H(C_b^k(M))$.

The next proposition can be proved in a similar way to the Riemannian case @GaJaRa.

\[countable:neigh\] Let $M$ be a complete $C^{k}$ Finsler manifold that is $C^k$ uniformly bumpable. Then, $\varphi \in H(C_b^k(M))$ has a countable neighborhood basis in $H(C_b^k(M))$ if and only if $\varphi\in M$.

A Myers-Nakai Theorem
=====================

Our main result is the following Banach-Stone type theorem for a certain class of Finsler manifolds. It states that the algebra structure of $C_b^k(M)$ determines the $C^k$ Finsler manifold. Recall that two normed algebras $(A,||\cdot||_A)$ and $(B,||\cdot||_B)$ are *equivalent as normed algebras* whenever there exists an algebra isomorphism $T:A\to B$ satisfying $||T(a)||_B=||a||_A$ for every $a\in A$. Let us begin by defining the class of Banach spaces where the Finsler manifolds shall be modeled.

A Banach space $(X,||\cdot||)$ is said to be **k-admissible** if for every equivalent norm $|\cdot|$ and $\varepsilon>0$, there are an open subset $B\supset \{x\in X: |x|\le 1\}$ of $X$ and a $C^k$ smooth function $g:B\to {\mathbb{R}}$ such that

1.  $|g(x)-|x||<\varepsilon$ for $x\in B$, and

2.  ${\operatorname{Lip}}(g)\le (1+\varepsilon)$ for the norm $|\cdot|$.

It is easy to prove the following lemma.

Let $X$ be a Banach space with one of the following properties:

1.  Density of the set of equivalent $C^k$ smooth norms: every equivalent norm on $X$ can be approximated in the Hausdorff metric by equivalent $C^k$ smooth norms @DGZ.

2.  $C^k$-fine approximation property ($k\ge 2$) and density of the set of equivalent $C^1$ smooth norms: For every $C^1$ smooth function $f:X\rightarrow \mathbb R$ and every $\varepsilon>0$, there is a $C^k$ smooth function $g:X\rightarrow \mathbb R$ satisfying $|f(x)-g(x)|<\varepsilon$ and $||f'(x)-g'(x)||<\varepsilon$ for all $x\in X$ (see @HJ, @Azfrygiljarlovo and @Moulis); also, every equivalent norm defined on $X$ can be approximated in the Hausdorff metric by equivalent $C^1$ smooth norms (see @DGZ [Theorem II 4.1]).

Then $X$ is k-admissible.

Banach spaces satisfying condition (A.2) are, for instance, separable Banach spaces with a Lipschitz $C^k$ smooth bump function. Banach spaces satisfying condition (A.1) for $k=1$ are, for instance, Weakly Compactly Generated (WCG) Banach spaces with a $C^1$ smooth bump function.

\[Myers:Nakai\] Let $M$ and $N$ be complete $C^k$ Finsler manifolds that are $C^k$ uniformly bumpable and are modeled on $k$-admissible Banach spaces. Then $M$ and $N$ are weakly $C^k$ equivalent as Finsler manifolds if and only if $C_b^k(M)$ and $C_b^k(N)$ are equivalent as normed algebras. Moreover, every normed algebra isomorphism $T:C_b^k(N)\to C_b^k(M)$ is of the form $T(f)=f\circ h$ where $h:M\to N$ is a weak $C^k$ Finsler isometry. In particular, $h$ is a $C^{k-1}$ Finsler isometry whenever $k\ge 2$.

In order to prove Theorem \[Myers:Nakai\], we shall follow the ideas of the Riemmanian case @GaJaRa. Let us divide the proof into several propositions.

\[isom\] Let $M$ and $N$ be $C^k$ Finsler manifolds such that $N$ is modeled on a $k$-admissible Banach space $Y$. Let $h:M\to N$ be a map such that $T:C_b^k(N)\to C_b^k(M)$ given by $T(f)=f\circ h$ is continuous. Then $h$ is $||T||$-Lispchitz for the Finsler metrics.

For every $y\in N$, let us take a chart $\psi_y: V_y\to  Y$ with $\psi_y(y)=0$. Let us consider the equivalent norm on $Y$, $|||\cdot|||_y:=||d\psi_y^{-1}(0)(\cdot)||_y$ and fix $\varepsilon>0$. Let us define the ball $B_{|||\cdot|||_y}(z,t):=\{w\in Y: \,
 |||w-z|||_y<t\}$.

**Fact.** For every $r>0$ such that $ B_{|||\cdot|||_y}(0,r)\subset \psi_y(V_y)$ and every $\widetilde{\varepsilon}>0$, there exists a $C^k$ smooth and Lipschitz function $f_y:Y\toÊ{\mathbb{R}}$ such that

1.  $f_y(0)=r$,

2.  $||f_y||_\infty:=\sup \{|f_y(z)|: \,z\in Y\} = r$,

3.  ${\operatorname{Lip}}(f_y)\le (1+\varepsilon)^2$ for the norm $|||\cdot|||_y$,

4.  $f_y(z)=0$ for every $z \in Y$ with $|||z|||_y\ge r$, and

5.  $|||z|||_y\le r-f_y(z)+\widetilde{\varepsilon}$ for every $|||z|||_y\le {r}$.

Let us prove the Fact. First of all, let us take $r>0$, $\widetilde{\varepsilon}>0$ and $0<\alpha< \min\{1,\frac{\varepsilon}{4},\frac{2\widetilde{\varepsilon}}{5r}\}$. Since $N$ is a $C^{k}$ Finsler manifold modeled on a $k$-admissible Banach space $Y$, there are an open subset $B\supset \{x\in Y: |||x|||_y\le  1\}$ of $Y$ and a $C^k$ smooth function $g:B\to {\mathbb{R}}$ such that

-   $|g(x)-|||x|||_y|<\alpha/2$ on $B$, and

-   ${\operatorname{Lip}}(g)\le (1+\alpha/2)$ for the norm $|||\cdot|||_y$.

Now, let us take a $C^\infty$ smooth and Lipschitz function $\theta:{\mathbb{R}}\to [0,1]$ such that

-   $\theta(t)=0$ whenever $t\le \alpha$,

-   $\theta(t)=1$ whenever $t\ge 1-\alpha$,

-   ${\operatorname{Lip}}(\theta)\le (1+\varepsilon)$, and

-   $|\theta(t)-t|\le 2\alpha$ for every $t\in[0,1+\alpha]$.

Let us define $$f(x)=
\begin{cases}
\theta(g(x)) & \text{if $x\in B$},\\
1 & \text{if $x\in Y\setminus B.$}
\end{cases}$$ It is straightforward to verify that $f$ is well-defined, $C^k$ smooth, $f(x)=1$ whenever $|||x|||_y \ge 1$ and $f(x)=0$ whenever $|||x|||_y \le \alpha/2$ Let us now consider $f_y:Y\to [0,r]$ as $f_y(z)=r(1-f(\frac{z}{r}))$, which is $C^k$ smooth, Lipschitz and satisfies:

-   $f_y(0)=r$,

-   $||f_y||_\infty = r$,

-   $|f_y(z)-f_y(x)|\le (1+\varepsilon)(1+\alpha/2)|||z-x|||_y\le (1+\varepsilon)^2|||z-x|||_y$,

-   $f_y(z)=0$ for every $z\in Y$ with $|||z|||_y\ge r$,

-   $|||\frac{z}{r}|||_y\le \frac{\alpha}{2} + g(\frac{z}{r})\le \frac{\alpha}{2}+2\alpha+f(\frac{z}{r})$ for every $|||z|||_y\le r$. Thus, $|||z|||_y\le r(\frac{\alpha}{2}+2\alpha) +r-f_y(z)\le \widetilde{\varepsilon}+r-f_y(z)$ for every $|||z|||_y\le {r}$.

Let us now prove Proposition \[isom\]. Let us fix $p_1,p_2\in M$ and $\varepsilon>0$. Let us consider $\sigma:[0,1]\to M$ a piecewise $C^1$ smooth path in $M$ joining $p_1$ and $p_2$, with $\ell(\sigma)\le d_M(p_1,p_2)+\varepsilon$. Since $h:M\to N$ is continuous, the path $\widehat{\sigma}:=h\circ \sigma: [0,1]\to N$, joining $h(p_1)$ and $h(p_2)$, is continuous as well. For every $q\in \widehat{\sigma}([0,1])$, there is $0<r_q<1$ and a chart $\psi_q:V_q\to Y$ such that $\psi_q(q)=0$, $B_N(q,r_{q})\subset V_q$ and the bijection $\psi_q:V_q\to \psi_q(V_q)$ is $(1+\varepsilon)$-bi-Lipschitz for the norm $||d\psi^{-1}_q(0)(\cdot)||_q$ in $Y$ (see Lemma \[desigualdades:BiLipschitz\]). Since $\widehat{\sigma}([0,1])$ is a compact set of $N$, there is a finite family of points $0=t_1<t_2<...<t_m=1$ and a family of open intervals $\{I_{k}\}_{k=1}^m$ covering the interval $[0,1]$ so that, if we define $q_k:=\widehat{\sigma}(t_k)$ and $r_k:=r_{q_k}$, for every $k=1,...,m$, we have

1.  $\widehat{\sigma}(I_{k})\subset B_N(q_k, r_{k}/(1+\varepsilon))$,

2.  $I_{j}\cap I_{k}\neq \emptyset$ if, and only if, $|j-k|\le 1$.

It is clear that $\widehat{\sigma}([0,1])\subset \bigcup_{k=1}^{m} B_N(q_k,\frac{r_k}{1+\varepsilon})$. Now, let us select a point $s_k\in I_k\cap I_{k+1}$ such that $t_k<s_k<t_{k+1}$, for every $k=1,...,m-1$. Let us write $a_k:=\widehat{\sigma}(s_k)$, for every $k=1,\cdots, m-1$, $\psi_k:=\psi_{q_k}$, $V_k:=V_{q_k}$ and $|||\cdot|||_k:=||d\psi^{-1}_{k}(0)(\cdot)||_{q_k}$, for every $k=1,....m$. Notice that $a_k\in  B_N(q_k, \frac{r_k}{1+\varepsilon})\cap B_N(q_{k+1}, \frac{r_{k+1}}{1+\varepsilon})$, for every $k=1, \cdots, m-1$. Since $\psi_{k}:V_{k}\to \psi_{k}(V_{k})$ is $(1+\varepsilon)$-bi-Lipschitz for the norm $|||\cdot|||_{k}$ in $Y$, we deduce that $\psi_{k}(a_k)\in B_{|||\cdot|||_{k}}(0,r_{k})$, for every $k=1, \cdots, m-1$.

Now, let us we apply the above Fact to $r_k$, $\varepsilon$ and $\widetilde{\varepsilon}=\varepsilon/2m$ to obtain functions $f_{k}:Y\to [0,r_k]$ satisfying properties (1)–(5), $k=1,\cdots, m$. Let us define the $C^k$ smooth and Lipschitz functions $g_k:N \to [0,r_k]$ as $g_k(z)=f_{k}(\psi_k(z))$ for every $z\in V_k$ and $g_k(z)=0$ for $z\not\in V_k$, $k=1,\cdots,m$. Then,

1.  $g_k\in C_b^k(N)$;

2.  $g_k(q_k)=r_{k}$;

3.  $|g_k(z)-g_k(x)|\le (1+\varepsilon)^3 d_N(z,x)$ for all $z,x\in N$;

4.  If $z\in \psi_k^{-1}(B_{|||\cdot|||_k}(0,{r_k}))$, then $|||\psi_k(z)|||_{k}\le {r_k}$ and from condition (5) on the Fact, we obtain $$d_N(z,q_k)\le (1+\varepsilon)|||\psi_k(z)-\psi_k(q_k)|||_{k}  = (1+\varepsilon)|||\psi_k(z)|||_{k} \le (1+\varepsilon)(r_{k}-g_k(z)+\varepsilon/2m).$$

The Lipschitz constant of $g_k\circ h$, for $k=1,\cdots, m$, is the following $$\begin{aligned}
{\operatorname{Lip}}(g_k\circ h) & \le ||g_k\circ h||_{C_b^1(M)}=||T(g_k)||_{C_b^1(M)} \le ||T|| ||g_k||_{C_b^1(N)}= \\
& = ||T|| \max\{||g_k||_\infty, ||dg_k||_\infty\}\le ||T|| (1+\varepsilon)^3.\end{aligned}$$ Now, since $r_{k}=g_{k}(q_k)=g_{k}(h(\sigma(t_k)))$ and $\psi_{k}(h(\sigma(s_k)))\in B_{|||\cdot|||_{k}}(0,{r_k})$, we have $$\begin{aligned}
d_N(h(p_1),&h(p_2))   \le \sum_{k=1}^{m-1} [d_N(h(\sigma(t_k)), h(\sigma(s_k))) +d_N(h(\sigma(s_k)), h(\sigma(t_{k+1}))) ] \le \\
&  \le \sum_{k=1}^{m-1} (1+\varepsilon)[ g_{k}(q_k)-g_{k}(h(\sigma(s_k)))+\\
&\qquad \qquad \quad \qquad \qquad \qquad \qquad + g_{k+1}(q_{k+1})-g_{k+1}(h(\sigma(s_k))) + \varepsilon/m ] \le \\
& \le \sum_{k=1}^{m-1} (1+\varepsilon)[{\operatorname{Lip}}(g_{k}\circ h) d_M(\sigma(t_k),\sigma(s_k)) + \\
& \qquad \qquad\qquad\qquad \qquad \qquad + {\operatorname{Lip}}(g_{k+1}\circ h)d_M(\sigma(t_{k+1}),\sigma(s_k)) +\varepsilon/m] \le \\
& \le \sum_{k=1}^{m-1} ||T||(1+\varepsilon)^4[d_M(\sigma(t_k),\sigma(s_k)) + d_M(\sigma(t_{k+1}),\sigma(s_k))] +\varepsilon(1+\varepsilon) \le \\
& \le \sum_{k=1}^{m-1} ||T||(1+\varepsilon)^4 \ell(\sigma_{\mid_{[t_k,t_{k+1}]}})+\varepsilon(1+\varepsilon) = ||T||(1+\varepsilon)^4 \ell(\sigma) +\varepsilon(1+\varepsilon) \le \\
&  \qquad \qquad\qquad \qquad \qquad  \qquad \qquad \le  ||T||(1+\varepsilon)^4(d_M(p_1,p_2)+\varepsilon)+\varepsilon(1+\varepsilon)\end{aligned}$$ for every $\varepsilon>0$. Thus, $h$ is $||T||$-Lipschitz.

\[weak:smooth\] Let $M$ and $N$ be $C^k$ Finsler manifolds such that $N$ is modeled on a Banach space with a Lipschitz $C^k$ smooth bump function. Let $h:M\to N$ be a homeomorphism such that $f\circ h \in C_b^k(M)$ for every $f\in C_b^k(N)$. Then, $h$ is a weakly $C^k$ smooth function on $M$.

Let us fix $x\in M$ and $\varepsilon=1$. There are charts $\varphi:U\to X$ of $M$ at $x$ and $\psi: V\to Y$ of $N$ at $h(x)$ satisfying inequalities and on $U$ and $V$, respectively. We can assume that $h(U)\subset V$. Since $Y$ admits a Lipschitz and $C^k$ smooth bump function and $\psi(h(U))$ is an open neighborhood of $\psi(h(x))$ in $Y$, there are real numbers $0<s<r$ such that $B(\psi(h(x)),s) \subset B(\psi(h(x)),r) \subset \psi(h(U))$ and a Lipschitz and $C^k$ smooth function $\alpha:Y\to {\mathbb{R}}$ such that $\alpha(y)=1$ for $y\in B(\psi(h(x)),s)$ and $\alpha(y)=0$ for $y\not\in B(\psi(h(x)),r)$. Let us define $U_0:=h^{-1}(\psi^{-1}(B(\psi(h(x)),s)))\subset U$, which is an open neighborhood of $x$ in $M$.

Let us check that $y^* \circ (\psi \circ  h \circ \varphi^{-1})$ is $C^k$ smooth on $\varphi(U_0)\subset X$ for all $y^* \in Y^*$. Following the proof of @GAJAANG [Theorem 4], we define $g:N\to {\mathbb{R}}$ as $g(y)=0$ whenever $y\not\in V$ and $g(y)=\alpha(\psi(y))\cdot y^*(\psi(y))$ whenever $y\in V$. It is clear that $g\in C_b^k(N)$ and, by assumption, $g\circ h \in C_b^k(M)$. Now, it follows that $\psi(h(\varphi^{-1}(z)))\in B(\psi(h(x)),s)$ for every $z\in \varphi(U_0)$. Thus $$\begin{aligned}
y^*\circ (\psi \circ  h \circ \varphi^{-1})(z) & = y^* (\psi(h(\varphi^{-1}(z))))=\alpha(\psi(h(\varphi^{-1}(z)))) y^*(\psi(h(\varphi^{-1}(z))))= \\
& =g(h(\varphi^{-1}(z)))=g\circ h\circ \varphi^{-1} (z),\end{aligned}$$ for every $z\in \varphi(U_0)$. Since $(g\circ h)\circ \varphi^{-1}$ is $C^k$ smooth on $\varphi(U_0)$, we have that $y^*\circ (\psi \circ  h \circ \varphi^{-1})$ is $C^k$ smooth on $\varphi(U_0)$. Thus $\psi \circ  h \circ \varphi^{-1}$ is weakly $C^k$ smooth on $\varphi(U_0)$ and $h$ is weakly $C^k$ smooth on $M$.

If $h:M\to N$ is a weak $C^k$ Finsler isometry, we can define the operator $T:C_b^k(N)\to C_b^k(M)$ by $T(f)=f\circ h$. Let us check that $T$ is well defined. For every $x\in M$, there are charts $\varphi:U\to X$ of $M$ at $x$ and $\psi:V\to Y$ of $N$ at $h(x)$, such that $h(U)\subset V$ and $\psi\circ h \circ \varphi^{-1}$ is weakly $C^k$ smooth on $\varphi(U)\subset X$. Also, $f\circ \psi^{-1}$ is $C^k$ smooth on $\psi(V)\subset Y$. Thus, by @GULLA [Proposition 4.2], $(f\circ \psi^{-1})\circ (\psi \circ h \circ \varphi^{-1})=f \circ h \circ \varphi^{-1}$ is $C^k$ smooth on $\varphi(U)$. Therefore, $f\circ h $ is $C^k$ smooth on $U$. Since this holds for every $x\in M$, we deduce that $f\circ h $ is $C^k$ smooth on $M$. Moreover, $T$ is an algebra isomorphism with $||T(f)||_{C_b^1(M)}=||f\circ h||_{C_b^1(M)}=||f||_{C_b^1(N)}$ for every $f\in C_b^k(N)$.

Conversely, let $T: C_b^k(N)\to C_b^k(M)$ be a normed algebra isometry. Then, we can define the function $h:H(C_b^k(M))\to H(C_b^k(N))$ by $h(\varphi)=\varphi\circ T$ for every $\varphi \in H(C_b^k(M))$. The function $h$ is a bijection. Moreover, $h$ is an homeomorphism. Recall that we identify $x\in M$ with $\delta_x\in C_b^k(M)$. Thus, $h(x)=h(\delta_x)=\delta_x\circ T$. Since $h$ is an homeomorphism, by Proposition \[countable:neigh\], we obtain for every $p\in N$ a unique point $x\in M$ such that $h(\delta_x)=\delta_{p}$. Let us check that $T(f)=f\circ h$ for all $f\in C_b^k(N)$. Indeed, for every $x\in M$ and every $f\in C_b^k(N)$, $$T(f)(x)=\delta_x(T(f))=(\delta_x\circ T)(f)=h(\delta_x)(f)=\delta_{h(x)}(f)=f(h(x))=f\circ h (x).$$

Now, from Proposition \[isom\] and Lemma \[weak:smooth\] we deduce that $h$ is a weak $C^k$ Finsler isometry.

It is worth mentioning that, for Riemannian manifolds, every metric isometry is a $C^1$ Finsler isometry. This result was proved by S. Myers and N. Steenrod @MyStee in the finite-dimensional case and by I. Garrido, J.A. Jaramillo and Y.C. Rangel @GaJaRa in the general case. Also, S. Deng and Z. Hou @DengHou obtained a version for finite-dimensional Riemannian-Finsler manifolds. Nevertheless, there is no a generalization, up to our knowledge, of the Myers-Steenrod theorem for all Finsler manifolds. Thus, for $k=1$ we can only assure that the metric isometry obtained in Theorem \[Myers:Nakai\] is weakly $C^1$ smooth.

Let us finish this note with some interesting corollaries of Theorem \[Myers:Nakai\]. First, recall that every separable Banach space with a Lipschitz $C^k$ smooth bump function satisfies condition (A.2) and every WCG Banach space with a $C^1$ smooth bump function satisfies condition (A.1) for $k=1$.

Let $M$ and $N$ be complete, $C^{1}$ Finsler manifolds that are $C^1$ uniformly bumpable and are modeled on WCG Banach spaces. Then $M$ and $N$ are weakly $C^1$ equivalent as Finsler manifolds if, and only if, $C_b^1(M)$ and $C_b^1(N)$ are equivalent as normed algebras. Moreover, every normed algebra isomorphism $T:C_b^1(N)\to C_b^1(M)$ is of the form $T(f)=f\circ h$ where $h:M\to N$ is a weak $C^1$ Finsler isometry.

Let $M$ and $N$ be complete, separable $C^{k}$ Finsler manifolds that are modeled on Banach spaces with a Lipschitz and $C^k$ smooth bump function. Then $M$ and $N$ are weakly $C^k$ equivalent as Finsler manifolds if and only if $C_b^k(M)$ and $C_b^k(N)$ are equivalent as normed algebras. Moreover, every normed algebra isomorphism $T:C_b^k(N)\to C_b^k(M)$ is of the form $T(f)=f\circ h$ where $h:M\to N$ is a weak $C^k$ Finsler isometry. In particular, $h$ is a $C^{k-1} $ Finsler isometry whenever $k\ge 2$.

Since every weakly $C^k$ smooth function with values in a finite-dimensional normed space is $C^k$ smooth and every finite-dimensional $C^{k}$ Finsler manifold is $C^k$ uniformly bumpable @MarLuis, we obtain the following Myers-Nakai result for finite-dimensional $C^{k}$ Finsler manifolds.

Let $M$ and $N$ be complete and finite dimensional $C^{k}$ Finsler manifolds. Then $M$ and $N$ are $C^k$ equivalent as Finsler manifolds if, and only if, $C_b^k(M)$ and $C_b^k(N)$ are equivalent as normed algebras. Moreover, every normed algebra isomorphism $T:C_b^k(N)\to C_b^k(M)$ is of the form $T(f)=f\circ h$ where $h:M\to N$ is a $C^k$ Finsler isometry.

We obtain an interesting application of Finsler manifolds to Banach spaces. Recall the well-known Mazur-Ulam Theorem establishing that every surjective isometry between two Banach spaces is affine.

Let $X$ and $Y$ be WCG Banach spaces with $C^1$ smooth bump functions. Then $X$ and $Y$ are isometric if, and only if, $C_b^1(X)$ and $C_b^1(Y)$ are equivalent as normed algebras. Moreover, every normed algebra isomorphism $T:C_b^1(Y)\to C_b^1(X)$ is of the form $T(f)=f\circ h$ where $h:X\to Y$ is a surjective isometry. In particular, $h$ and $h^{-1}$ are affine isometries.

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[^1]: Supported in part by DGES (Spain) Project MTM2009-07848. L. Sánchez-González has also been supported by grant MEC AP2007-00868

{'source': 'arXiv_out/1108'}