A note on the dimension of isometry group of a Riemannian manifold

Abstract

In this paper, we obtain some results on the dimension of the isometry group of a Riemannian manifold. In specific dimensions, we give a range which the dimension of an isometry group can not be in. We also give necessary conditions for a manifold to have a free canonical action on some specific manifolds. We give a boundary of the dimension of the full isometry group if the dimension of a manifold is greater or equal to 4.