On Infinitesimal Holonomy and Isotropy Groups

1 February 1957

journal article
research article
Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal

Vol. 11, 111-114
https://doi.org/10.1017/s0027763000001987

Abstract

We have proved in [2] that if the restricted homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at every point, then the Riemannian manifold is locally symmetric, that is, the covariant derivatives of the curvature tensor field are zero. The proof of this theorem, however, depended on an insufficiently stated proposition (Theorem 1, [2]). In the present note, we shall give a proof of a more general theorem of the same type.

Keywords

This publication has 3 references indexed in Scilit:

Reduction Theorem for Connections and its Application to the Problem of Isotropy and Holonomy Groups of a Riemannian Manifold
Nagoya Mathematical Journal, 1955
Invariant Affine Connections on Homogeneous Spaces
American Journal of Mathematics, 1954
On the Holonomy Groups of Linear Connections*)*)This work was partially supported by an Office of Naval Research contract at Princeton University.
Indagationes Mathematicae, 1953