On the differentiability of isometries

Abstract

Let 9TC be an re-dimensional differentiable manifold of class Ck, k = l, • • • , oo, w (as usual O means analytic, and we use the conventions * ±m= and w + w=w for integers m). An automorphism of 9TC is a one-to-one map of 911 onto itself which is of class C* and is nonsingular. Now let 9TC be given a Riemannian structure, i.e. a positive definite covariant tensor field of rank two, and suppose that in some family of admissible coordinate systems that cover 9TC the components of this tensor field are of class Ck+1. This hypothesis will be satisfied, for example, if 911 is the manifold of class Ck associated with a manifold 9TC of class Ck+2 and the metric tensor field is a tensor field of highest possible class (namely Ck+1) on 911. Moreover this hypothesis implies that in the neighborhood of each point of 9TC we can define a Riemannian normal coordinate system and that such a coordinate system is admissible, i.e. of class C*. The Riemannian structure on 9H gives rise in a well-known way to a metric p on the underlying point set of 9ft, and we denote by M the metric space that results. An automorphism of M is of course a oneto-one map of M onto itself which preserves distance, i.e. an isometry. A remarkable result of Myers and Steenrod [l] states that an automorphism of M is always an automorphism of 9TC. It is natural to conjecture that the following stronger result is true.