Algebras of differentiable functions on Riemannian manifolds

Abstract

For an infinite-dimensional Riemannian manifold M we denote by C b 1 ( 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 C b 1 ( 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.