E extracta mathematicae Vol. 25, Núm. 3, 249 – 261 (2010) Lip-density and Algebras of Lipschitz Functions on Metric Spaces ∗ I. Garrido, J.A. Jaramillo, Y.C. Rangel Departamento de Geometŕıa y Topoloǵıa, Universidad Complutense de Madrid, 28040 Madrid, Spain, maigarri@mat.ucm.es Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain, jaramil@mat.ucm.es Departamento de Matemática, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela, yrangel25@hotmail.com Presented by Francisco Montalvo Received December 21, 2010 Abstract : Our aim in this note is to give an extension of the classical Myers-Nakai theorem in the context of Finsler manifolds. To achieve this, we provide a general result in this line for subalgebras of bounded Lipschitz functions on length metric spaces. We also establish some connection with the uniform approximation of bounded Lipschitz functions by functions in the subalgebra, keeping control on the Lipschitz constants. Key words: Algebras of Lipschitz functions, approximation, Finsler manifolds. AMS Subject Class. (2010): 54C35, 46E25, 58B20. 1. Introduction There is a wide range of results in the literature asserting that the topolog- ical, metric or differentiable structure of a given space X can be characterized in terms of a suitable algebraic or topological-algebraic structure on the space C(X) of continuous real functions on X, or on a certain subfamily of C(X). We should mention, as fundamental prototype, the classical Banach-Stone theorem, from which the topology of a compact space X is determined by the linear metric structure of C(X) (endowed with the sup-norm). We refer to [3] and references therein for further information about different extensions, generalizations and variants of this result. We are especially interested here in the theorem of Myers-Nakai, giving that the Riemannian structure of a Riemannian manifold M is determined by ∗ Isabel Garrido and Jesús A. Jaramillo have been supported in part by D.G.I. (Spain) Grant MTM2009-07848. Yenny Rangel has been associated to the Proyect 004-RCT-2010 (UCLA) (Venezuela). 249 250 i. garrido, j.a. jaramillo, y.c. rangel 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 thatM is compact, and later on by Nakai [12] for general (finite-dimensional) manifolds. The case of infinite-dimensional Riemannian manifolds is considered in [6]. Furthermore, by the Myers-Steenrod theorem (see [11]), the Riemannian structure of a connected Riemannian manifold is in turn determined by the purely metric structure of the manifold endowed with its geodesic distance. In [4] we have obtained a result of this type connecting the isometric type of a complete length metric space X with the normed algebra structure of the corresponding space Lip∗(X) of all bounded Lipschitz functions on X (see also [5] for related results in the class of the so-called small-determined metric spaces). Our first aim in this note is to give an extension of this result for certain subalgebras of bounded Lipschitz functions on length metric spaces. In order to achieve this, we consider the notion of Lip-density for a subalgebra A of Lip∗(X), which provides uniform approximation of an arbitrary function f ∈ Lip∗(X) by functions in A, in such a way that the Lipschitz constants of the approximating functions also approach the Lipschitz constant of f . Finally, as a consequence, we obtain an extension of the Myers-Nakai theorem to the context of Finsler manifolds. This note is based on the talk delivered by the second author at the “Sev- enth Italian-Spanish Conference in General Topology and its Applications” which took place in Badajoz (Spain), September 2010. It is a great pleasure to thank the organizers for their invitation to participate in the Conference. 2. Lip-density in metric spaces Let (X, d) be a metric space. We denote by Lip(X) the space of all real Lipschitz functions defined on X, and by Lip∗(X) the space of all bounded real Lipschitz functions defined on X. For f ∈ Lip(X), its Lipschitz constant is defined, as usual, by Lip(f) = sup { |f(x)− f(y)| d(x, y) : x, y ∈ X , x ̸= y } . The space Lip∗(X) is a complete normed algebra endowed with the natural norm: ∥f∥Lip∗(X) = ∥f∥∞ ∨ Lip(f) lip-density and algebras of lipschitz functions 251 where ∥ · ∥∞ denotes the sup-norm and ∨ denotes the supremum. Definition 1. Let (X, d) be a metric space. We say that a subset G ⊂ Lip(X) is Lip-dense in Lip(X) if for every f ∈ Lip(X) and every ε > 0 there exists some g ∈ G such that: (i) |f(x)− g(x)| ≤ ε for each x ∈ X, (ii) Lip(g) ≤ Lip(f) + ε. In the same way, we say that G ⊂ Lip∗(X) is Lip-dense in Lip∗(X) if for every f ∈ Lip∗(X) and every ε > 0 there exists some g ∈ G such that conditions (i) and (ii) above hold. The most remarkable examples of Lip-density are provided by subsets of smooth functions defined on a Riemannian manifold or, more generally, on a Finsler manifold. The notion of Finsler manifold can be traced back to Riemann in his 1854 Habilitationschrift, where he suggested the possibility of considering more general, non-quadratic, norms for measuring the length of an element of arc. The systematic study was initiated by P. Finsler in his thesis (1919). We refer to the book by Bao, Chern and Shen [1] for a detailed introduction to this subject. The basic idea of a Finsler structure on a smooth manifold is a smooth assignment, on each tangent space, of a so-called Minkowski norm, which is a kind of norm with nice properties of convexity and smoothness. More precisely: Definition 2. Let V be a finite-dimensional real vector space. We say that a functional F : V → [0,∞) is a Minkowski norm on V if it satisfies: (1) Positivity: F (v) = 0 if, and only if, v = 0. (2) Triangle inequality: F (u+ v) ≤ F (u) + F (v), for every u, v ∈ V . (3) Positive homogeneity: F (λv) = λF (v), for every v ∈ V and every λ > 0. (4) Regularity: F is continuous on V and C∞ on V \ {0}. (5) Strong convexity: For every v ∈ V \{0}, the quadratic form gv associated to the second derivative of the function F 2 at v, that is gv = 1 2 d2 [ F 2 ] (v) , is positive-definite on V . 252 i. garrido, j.a. jaramillo, y.c. rangel We note that conditions (1) and (2) in the above definition are, in fact, consequence of conditions (3) – (5) (see [1, Theorem 1.2.2]). It is clear that every norm associated to an inner product is a Minkowski norm. We also recall that, in general, a Minkowski norm needs not be symmetric, and there are indeed very interesting examples of nonsymmetric Minkowski norms, such as for example Randers spaces (see [1]). We say F is symmetric or absolutely homogeneous if: F (λv) = |λ|F (v) for every v ∈ V and every λ ∈ R . In this case, F is a norm in the usual sense. Now the definition of Finsler manifold is as follows: Definition 3. 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, ·) : TxM → [0,∞) is a Minkowski norm on the tangent space TxM . In particular, a Riemannian manifold is a special case of Finsler manifold, where the Minkowski norm on each tangent space TxM 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 let (M,F ) be a Finsler manifold and 1 ≤ k ≤ ∞. If f : M → R is a Ck function, we define as usual the norm of its differential df(x) at a point x ∈ M by: ∥df(x)∥F = sup {|df(x)(v)| : v ∈ TxM , F (x, v) ≤ 1} . We denote by Ck b (M) the space of all real bounded Ck-functions defined on M whose derivative has uniformly bounded norm. We endow Ck b (M) with the natural norm: ∥f∥Ck b (M) = sup x∈M |f(x)| ∨ sup x∈M ∥df(x)∥F . Endowed with this norm, Ck b (M) is a normed algebra, and in particular C1 b (M) is a complete normed algebra. lip-density and algebras of lipschitz functions 253 Now suppose that (M,F ) is a connected Finsler manifold. The Finsler distance dF on M is defined by: dF (x, y) = inf { ℓF (σ) : σ piecewise C1 path from x to y}, where the Finsler length of a piecewise C1 path σ : [a, b] → M is defined as: ℓF (σ) = ∫ b a F (σ(t), σ′(t)) dt. In this way we have (see Section 6.2 of [1]) that the Finsler distance dF is an asymmetric distance on M , in the sense that it verifies: • dF (x, y) ≥ 0; • dF (x, y) = 0 if, and only if, x = y; • dF (x, y) ≤ dF (x, z) + dF (z, y) for every x, y, z ∈ M . In general, dF needs not be symmetric. Nevertheless, if F is reversible the Finsler distance dF is symmetric, and therefore (M,dF ) is a metric space in the usual sense. In this case, as a consequence of the corresponding mean value theorem, we obtain in [7] the following: Lemma 4. Let (M,F ) be a connected, reversible Finsler manifold, and consider the associated Finsler distance dF on M . For each C1 function f : M → R we have that f : (M,dF ) → R is Lipschitz if, and only if, f has bounded derivative. In fact, Lip(f) = sup x∈M ∥df(x)∥F . In particular, from the above lemma we deduce that if (M,F ) is a con- nected, reversible Finsler manifold, then Ck(M) ∩ Lip∗(M) = Ck b (M). We recall now the following approximation result due to Greene and Wu (see [8, Proposition 2.1]): Theorem 5. ([8]) Let M be a connected and second countable Rieman- nian manifold, endowed with its Riemannian distance. Then C∞(M)∩Lip(M) is Lip-dense in Lip(M), and therefore C∞ b (M) is Lip-dense in Lip∗(M). In the case of Finsler manifolds, we do not know whether it is possible to obtain in general Lip-approximation of Lipschitz functions by C∞-smooth 254 i. garrido, j.a. jaramillo, y.c. rangel functions. Nevertheless, we have the following result from [7] giving Lip- approximation by C1-smooth functions. In the proof, we use the exponential map associated to the Finsler structure, which in general is only C1-smooth. There is a relevant class of Finsler manifolds which are characterized by the fact that the exponential map is C∞-smooth. These are called Berwald spaces, and include, of course, the Riemannian manifolds (see [1, Section 5.3]). In the class of Berwald spaces, the same proof gives Lip-approximation by C∞- smooth functions. Theorem 6. ([7]) LetM be a connected, reversible and second countable Finsler manifold, endowed with its Finsler distance. Then C1(M) ∩ Lip(M) is Lip-dense in Lip(M), and therefore C1 b (M) is Lip-dense in Lip∗(M). We also note that the above approximation theorem is proved in [7] in the general case of non-reversible Finsler manifolds, using a suitable definition of Lipschitz functions in this context. To finish this section we are going to show that Lip-density corresponds with the usual notion of density for a suitable metric on Lip(X). Namely, the metric defined by ρ(f, g) = sup x∈X {|f(x)− g(x)| ∧ 1} ∨ |Lip(f)− Lip(g)|, where ∨ and ∧ denotes the sup and inf, respectively. Indeed, it is clear that density for the metric ρ implies Lip-density. Conversely, let G be Lip-dense in Lip(X) and f ∈ Lip(X). Given 0 < ε < 1, choose x0 ̸= y0 in X such that Lip(f)− ε 2 < |f(x0)− f(y0)| d(x0, y0) . Now, take 0 < ε′ < inf{ε, (14 ε) d(x0, y0)}. By Lip-density we can find g ∈ G such that |f(x)− g(x)| ≤ ε′ for each x ∈ X, and Lip(g) ≤ Lip(f) + ε′. Then, Lip(g) ≥ |g(x0)− g(y0)| d(x0, y0) ≥ |f(x0)− f(y0)| − 2ε′ d(x0, y0) ≥ Lip(f)− ε 2 − 2ε′ d(x0, y0) > Lip(f)− ε . And, therefore ρ(f, g) < ε. Moreover, note that on Lip∗(X) the norm ∥ · ∥Lip∗ induces a topology finer than the metric topology given by ρ. In general these topologies are not equiv- alent. For instance, if M is a connected and second countable Riemannian (or, lip-density and algebras of lipschitz functions 255 more generally, Finsler) manifold, then C1 b (M) is ∥·∥Lip∗-closed but Lip-dense in Lip∗(M). 3. The structure space In this section we consider a metric space (X, d) and a unital and Lip- dense subalgebra A of Lip∗(X). We are going to recall the construction of the structure space associated to A, following the lines of Isbell in [9]. First of all note that, by Lip-density, A separates points and closed sets of X, and this implies in particular that X is endowed with the weak topology given by A. As usual, we say that φ : A → R is an algebra homomorphism whenever it satisfies: φ(λf + µg) = λφ(f) + µφ(g) , φ(f · g) = φ(f) · φ(g) , for all f, g ∈ A and for all λ, µ ∈ R. Note that an algebra homomorphism φ is nonzero if, and only if, φ(1) = 1. On the other hand, we say that φ is positive if φ(f) ≥ 0 whenever f ≥ 0. We define the structure space M(A) as the set of all nonzero and positive algebra homomorphisms φ : A → R, considered as a topological subspace of the product RA. It is not difficult to see that M(A) is closed in RA. Moreover, since every function in A is bounded, M(A) is in particular a compact space. Now, consider the natural map δ : X → M(A) given by δ(x) = δx, where δx is the point evaluation homomorphism, i.e., δx(f) = f(x), for every f ∈ A. Clearly, δ is a continuous map. On the other hand, the subspace δ(X) is dense in M(A). Indeed, given φ ∈ M(A), f1, . . . , fn ∈ A, and ε > 0, there exists some x ∈ X such that |δx(fi)− φ(fi)| < ε, for all i = 1, . . . , n. Otherwise the function g = ∑n i=1(fi − φ(fi)) 2 ∈ A would satisfy g ≥ ε2 and φ(g) = 0, and this is impossible since φ is positive. Finally, from the fact that A separates points and closed sets of X, we can derive that δ is a topological embedding, and therefore M(A) can be considered as a compactification of X. In addition this compactification has the property that each f ∈ A admits a continuous extension f̂ to M(A), namely by defining f̂(φ) = φ(f), for all φ ∈ M(A). Note that this extension f̂ coincides on M(A) with the corresponding projection map πf : RA → R. Next we will see that, thanks to Lip-density, the points in X can be topo- logically distinguished into M(A). 256 i. garrido, j.a. jaramillo, y.c. rangel Lemma 7. Let (X, d) be a complete metric space, and A a unital and Lip- dense subalgebra of Lip∗(X). Then φ ∈ M(A) has a countable neighborhood basis in M(A) if, and only if, φ ∈ X. Proof. Assume first that φ ∈ M(A) \ X has a countable neighborhood basis. Since X is dense in M(A) there exists a sequence (xn) in X converging to φ. From the completeness of X it follows that (xn) has no Cauchy subse- quence, and therefore there exist some ε > 0 and a subsequence (xnk ) of (xn) such that d(xnk , xnj ) ≥ ε for k ̸= j. Consider the sets P = {xn2k : k ∈ N} and Q = {xn2k+1 : k ∈ N}, and define the function f(x) = min{ε, d(x, P )}. It is clear that f ∈ Lip∗(X) and we have that f(x) = 0 for every x ∈ P and f(x) ≥ ε for every x ∈ Q. By Lip-density, there exists g ∈ A such that |f(x)− g(x)| ≤ ε 4 , for each x ∈ X. Therefore the extended function ĝ defined on the whole space M(A) takes values ≤ ε 4 on clM(A)P and values ≥ 3ε 4 on clM(A)Q (where clM(A) denotes the closure operator on the space M(A)). But this is a contradiction since φ ∈ clM(A)P ∩ clM(A)Q. Conversely, if φ ∈ X, consider Bn the open ball in X with centre φ and radius 1/n. Then the family {clM(A)Bn} is easily seen to be a countable neighborhood basis as required. Note that along this section it would not be necessary to suppose that A is Lip-dense, but only uniformly dense. 4. Isometries between length spaces Recall that a metric space (X, d) is said to be a length space if for every x, y ∈ X d(x, y) = inf {ℓ(γ) : γ is a continuous path from x to y }. Here the length of a continuous path γ : [a, b] → X is defined, as usual, by ℓ(γ) = sup { k∑ i=1 d(γ(ti), γ(ti−1)) } ∈ [0,+∞] where a = t0 < t1 < · · · < tk = b runs over all partitions of the interval [a, b]. Definition 8. Let (X, dX) and (Y, dY ) be metric spaces and consider two subsets AX ⊂ Lip∗(X) and AY ⊂ Lip∗(Y ), respectively. We say that a bijection h : Y → X is stable for AX and AY if, for every function f : X → R, we have that f ∈ AX if, and only if, f ◦ h ∈ AY . lip-density and algebras of lipschitz functions 257 The next result shows in particular that if a bijection h : Y → X between length metric spaces is stable for some Lip-dense subalgebras, then h is in fact a bi-Lipschitz mapping. Lemma 9. Let (X, dX) and (Y, dY ) be length metric spaces, h : Y → X a mapping, and consider two Lip-dense subalgebras AX ⊂ Lip∗(X) and AY ⊂ Lip∗(Y ), respectively. Suppose that T : AX → AY is a continuous homomorphism for the corresponding Lip∗-norms given by T (f) = f ◦ h, for every f ∈ AX . Then h is ∥T∥-Lipschitz. Proof. Firstly, note that h is continuous. Indeed, as we said before, the Lip-density of AX in Lip∗(X) implies that X is endowed with the weak topol- ogy given by AX , and therefore h is continuous because f ◦h is continuous on Y , for every f ∈ AX . Now, in order to see that h is ∥T∥-Lipschitz, let p, q ∈ Y , and suppose first that dX(h(p), h(q)) ≤ 1. Fix ε > 0, and consider the function f : X → R, defined by f(x) = min{1, dX(h(p), x)}. It is clear that f is 1-Lipschitz and 0 ≤ f ≤ 1, so by Lip-density there exists g ∈ AX such that (1) |f(x)− g(x)| ≤ ε for every x ∈ X, (2) Lip(g) ≤ Lip(f) + ε = 1 + ε. Since, in particular, |g(x)| ≤ f(x) + ε ≤ 1 + ε, for every x ∈ X, then we have that ∥g∥Lip∗(X) ≤ 1 + ε. Now, the continuity of T gives that: Lip(g ◦ h) ≤ ∥g ◦ h∥Lip∗(Y ) ≤ ∥T∥ · ∥g∥Lip∗(X) ≤ (1 + ε) · ∥T∥ . Therefore, dX(h(p), h(q)) = f(h(q)) = |f(h(q))− f(h(p))| ≤ |g(h(q))− g(h(p))| + 2ε ≤ Lip(g ◦ h) · dY (p, q) + 2ε ≤ (1 + ε) · ∥T∥ · dY (p, q) + 2ε . Taking ε → 0 we obtain that dX(h(p), h(q)) ≤ ∥T∥ · dY (p, q). Next, given arbitrary p, q ∈ Y consider h(p), h(q) ∈ X. For every ε > 0, let σ : [a, b] → Y a continuous path from p to q with ℓ(σ) ≤ dY (p, q) + ε and let σ̂ = h ◦ σ. Consider a partition P = {a = t0 < t1 < · · · < tk = b} of the 258 i. garrido, j.a. jaramillo, y.c. rangel interval [a, b] such that dX(h(σ(ti−1)), h(σ(ti))) < 1, for each i = 1, 2, . . . , k. By the previous case, we have that dX(h(p), h(q)) ≤ k∑ i=1 dX ( h(σ(ti−1)), h(σ(ti)) ) ≤ k∑ i=1 ∥T∥ · dY ( σ(ti−1), σ(ti) ) ≤ k∑ i=1 ∥T∥ · ℓ ( σ|[ti−1,ti] ) = ∥T∥ · k∑ i=1 ℓ ( σ|[ti−1,ti] ) = ∥T∥ · ℓ(σ) ≤ ∥T∥ · ( dY (p, q) + ε ) . Taking ε → 0, we deduce that dX(h(p), h(q)) ≤ ∥T∥ · dY (p, q). We are going to establish a connection between isometries of length metric spaces which are stable for some Lip-dense subalgebras, and linear isome- tries between the corresponding subalgebras. We first recall the following definition. Definition 10. Let (A, ∥ · ∥A) and (B, ∥ · ∥B) be normed algebras. We say that T : A → B is an isometry of normed algebras if: (i) T is bijective, linear and multiplicative. (ii) ∥T (f)∥B = ∥f∥A, for every f ∈ A. Now we give our main result: Theorem 11. Let (X, dX) and (Y, dY ) be complete length metric spaces, and consider two Lip-dense subalgebras AX ⊂ Lip∗(X) and AY ⊂ Lip∗(Y ), respectively. Then the following are equivalent: (a) The mapping T : ( AX , ∥ · ∥Lip∗(X) ) → ( AY , ∥ · ∥Lip∗(Y ) ) is an isometry of normed algebras. (b) There exists an isometry h : Y → X stable for AX and AY , such that T (f) = f ◦ h, for every f ∈ AX . lip-density and algebras of lipschitz functions 259 In particular, if AX and AY are isometric as normed algebras, then X and Y are isometric. Proof. Firstly, suppose that T : AX → AY is an isometry of normed alge- bras. Since T (1) = T (1·1) = T (1)2, then the function T (1) ∈ AY can only take the values 0 and 1. Taking into account that Y is path-connected, we deduce that T (1) is a constant function, and since ∥T (1)∥Lip∗(Y ) = ∥1∥Lip∗(X) = 1, we obtain that T (1) = 1. On the other hand, T must be positive. Otherwise, there would exist 0 ≤ f ∈ AX such that T (f) is not positive, i.e., there is some y0 ∈ Y with T (f)(y0) < 0. Take a real number R > ∥f∥Lip∗(X), and consider g = −f +R. Then 0 ≤ g ∈ AX and ∥g∥Lip∗(X) ≤ R. But, ∥T (g)∥Lip∗(Y ) = ∥ − T (f) +R∥Lip∗(Y ) ≥ ∥ − T (f) +R∥∞ ≥ | − T (f)(y0) +R| > R which would be a contradiction. In this way, we can consider the mapping h : M(AY ) → M(AX) between the corresponding structure spaces by defining h(φ) = φ◦T , for every φ ∈ AY . It is clear that h is a bijection, and in fact h is a homeomorphism since πf ◦h = πT (f), for every f ∈ AX , where πf and πT (f) denote the corresponding projection maps on the product spaces. By Lemma 7, we have that a point φ ∈ M(AX) has a countable neighborhood basis in M(AX) if, and only if, φ ∈ X, and the same is true for M(AY ). Therefore, h takes Y onto X. Also, for every f ∈ AX and for every y ∈ Y we have that T (f)(y) = δy(T (f)) = (δy ◦ T )(f) = h(δy)(f) = f(h(δy)) = (f ◦ h)(y) so it follows that T (f) = f ◦h. In particular, we have that f ◦h ∈ AY whenever f ∈ AX . Now by Lemma 9 we deduce that h : Y → X es 1-Lipschitz, since T is an isometry and ∥T∥ = 1. On the other hand, working with T−1 and h−1, we also obtain that h−1 ◦ g ∈ AX whenever f ∈ AY and h−1 is also 1-Lipschitz. In this way we deduce that h : Y → X is an isometry stable for AX and AY . Finally, it is easy to check that condition (b) implies (a). 260 i. garrido, j.a. jaramillo, y.c. rangel 5. A Myers-Nakai theorem In this section we obtain from our previous results a version of the classical Myers-Nakai theorem in the context of Finsler manifolds. Recall that a map- ping h : (M,F ) → (N,G) between Finsler manifolds is said to be a Finsler isometry if h is a diffeomorphism which preserves the Finsler structure, that is, for every x ∈ M and every v ∈ TxM : F (x, v) = G ( h(x),dh(x)(v) ) . A classical result due to Myers and Steenrod [11] asserts that, if M and N are connected Riemannian manifolds, a mapping h : M → N is a Riemannian isometry if, and only if, it is a metric isometry for the corresponding Rieman- nian distances. This result has been extended by Deng and Hou [2] to the context of Finsler manifolds: Theorem 12. ([2]) Let (M,F ) and (N,G) be connected Finsler mani- folds. Then h : M → N is a Finsler isometry if, and only if, h is bijective and preserves the corresponding Finsler distances, that is, for every x, y ∈ M : dF (x, y) = dG(h(x), h(y)) . Now combining Theorem 12 above with Theorem 11 and the approxima- tion result in Theorem 6, we obtain the following Myers-Nakai theorem for Finsler manifolds. Theorem 13. Let M and N be connected, reversible, complete and sec- ond countable Finsler manifolds. Then M and N are equivalent as Finsler manifolds if, and only if, C1 b (M) and C1 b (N) are equivalent as normed alge- bras. 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. References [1] D. Bao, S.S. Chern, Z. 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