A characterization of locally homogeneous Riemann manifolds of dimension 3

Abstract

It is classical to characterize locally homogeneous Riemann manifolds by infinitesimal conditions. For example, [Si] asserts that the local-homogeneity is equivalent to the existence of linear isometries between tangent spaces which preserve the curvatures and their covariant derivatives up to certain orders. It is also known that the local homogeneity is equivalent to the existence of a certain tensor field of type (1, 2) (for this and a further study, see [TV]).