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Algebras of differentiable functions on Riemannian manifolds

For an infinite‐dimensional Riemannian manifold M we denote by Cb1(M) the space of all real bounded functions of class C1 on M with bounded derivative. In this paper we shall see how the natural structure of normed algebra on Cb1(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.