Algebras of differentiable functions and algebras of Lipschitz functions (Q1063204)

The results of \textit{S. B. Myers} [Proc. Am. Math. Soc. 5, 917-922 (1954; Zbl 0057.095)] on compact differentiable manifolds and compact Riemannian manifolds, have been extended in this paper in an appropriate manner to non-compact differentiable manifolds and non-compact Riemannian manifolds. More precisely; it has been proved in this paper that if M is a non- compact \(C^ r\)-differentiable manifold, then the algebra \(C^ r_{00}(M)\) of differentiable functions of class \(C^ r\) with compact support uniquely determine to within diffeomorphism the manifold M. If in addition M is a Riemannian manifold, then \(C^ r_{00}(M)\) carries the structure of a normed algebra (not in general complete). Then it has been proved that the Riemannian manifold M is uniquely determined (again, to within diffeomorphism) by the normed algebra \(C^ r_{00}(M).\) In the last part of this paper it has been proved that if M is a compact connected Riemannian manifold of diameter (as a metric space) less than or equal to 2, then every isometry of the Banach space Lip M of Lipschitz functions on M is induced by an isometry of M.