The holonomy group of a locally symmetric space

TL;DR

This paper proves that the holonomy group of a connected Riemannian locally symmetric space, without flat factors, is compact and has finite index in its normalizer within the orthogonal group.

Contribution

It establishes a new structural property of holonomy groups in locally symmetric spaces, extending previous results to non-complete cases.

Findings

Holonomy group is compact for such spaces.

Holonomy group has finite index in its normalizer.

Results apply to non-complete locally symmetric spaces.

Abstract

We show that the holonomy group of a connected Riemannian locally symmetric space (not necessarily complete) without local flat factor is compact and has finite index in its normalizer in the orthogonal group.