Algebras of differentiable functions on Riemannian manifolds

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For an infinite-dimensional Riemannian manifold M we denote by C1b(M) the space of all real bounded functions of class C(1) on M with bounded derivative. In this paper we shall see how the natural structure of normed algebra on C1b(M) characterizes the Riemannian structure of M, for the special case of the so-called uniformly bumpable manifolds. For that we need, among other things, to extend the classical Myers-Steenrod theorem on the equivalence between metric and Riemannian isometries, to the setting of infinite-dimensional Riemannian manifolds.
D. Azagra, J. Ferrera and F. López-Mesas, ‘Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds’, J. Funct. Anal. 220 (2005) 304–361. D. Azagra, J. Ferrera, F. López-Mesas and Y. Rangel, ‘Smooth approximation of Lipschitz functions on Riemannian manifolds’, J. Math. Anal. Appl. 326 (2007) 1370–1378. M. I. Garrido and J. A. Jaramillo, ‘Variations on the Banach–Stone theorem’, Extracta Math. 17 (2002) 351–383. M. I. Garrido and J. A. Jaramillo, ‘Homomorphisms on function lattices’, Monatsh. Math. 141 (2004) 127–146. S. Helgason, Differential geometry, Lie groups, and symmetryc spaces (Academic Press, New York, 1978). J. R. Isbell, ‘Algebras of uniformly continuous functions’, Ann. of Math. (2) 69 (1958) 96–125. S. Lang, Fundaments of differential geometry, Graduate Text in Mathematics 191 (Springer, New York,1999). S. Mazur and S. Ulam, ‘Sur les transformations isométriques d’espaces vectoriels norm´es’, C. R. Math.Acad. Sci. Paris 194 (1932) 946–948. S. B. Myers, ‘Algebras of differentiable functions’, Proc. Amer. Math. Soc. 5 (1954) 917–922. S. B. Myers and N. E. Steenrod, ‘The group of isometries of a Riemannian manifold’, Ann. of Math.(2) 40 (1939) 400–416. M. Nakai, ‘Algebras of some differentiable functions on Riemannian manifolds’, Japan. J. Math. 29 (1959) 60–67.