The Group of Isometries of a Riemannian Manifold

In the classical theory of groups of motions (isometries) of a Riemannian space,1 neither whole groups nor whole spaces were ever considered, but only germs and spaces in the neighborhood of a point. Furthermore one assumed given a Lie group germ of motions and then deduced theorems concerning the structure of the space. The present paper contains a study of the total group of isometries of a Riemannian manifold in the large. The manifold M is taken of class Cr (r > 2); and by an isometry is meant any distance preserving transformation. A natural topology is defined in the group G of all isometrics. The motion of a point under the group is studied and is found to describe a submanifold of class C1. This leads to the introduction of parameters in G from which it follows that G is locally euclidean. It is then proved that, under these parameters, G is a Lie group of transformations of M. The same proposition is proved for any closed subgroup of G. Thus the classical theory of Lie groups of motions applies to any closed group of motions.