• AFFINE TRANSFORMATIONS AND ISOMETRIES IN A COMPLETE RIEMANNIAN MANIFOLD

Abstract

The groups of affine transformations of an affinely connected manifold were studied by Nomizu and also Hano-Morimato. Further, Myers and Steenrod gave the theory of group of isometries of Riemannian manifold. In the present study we describe certain aspects of a complete Riemannian manifold. We have investigated that in a complete irreducible Riemannian manifolds the group of all affine transformations and the group of all isometries are equal. Further, if X be an infinitesimal affine transformation on a complete Riemannian manifold M, then X is an infinitesimal isometry. Also, we have related the affine transformations on a complete Riemannian manifold to the conditions of isometry.