Sufficient conditions for a Riemannian manifold to be locally symmetric

Kouei Sekigawa and Shûkichi Tanno

Vol. 34 (1970), No. 1, 157–162

DOI: 10.2140/pjm.1970.34.157

Abstract

In a locally symmetric Riemannian manifold the scalar curvature is constant and each k-th covariant derivative of the Riemannian curvature tensor vanishes. In this note, we show that if the covariant derivatives of the Riemannian curvature tensor satisfy some algebraic conditions at each point, then the Riemannian manifold is locally symmetric.

Mathematical Subject Classification

Primary: 53.73

Milestones

Received: 6 October 1969

Published: 1 July 1970