Semisimplicity of indefinite extrinsic symmetric spaces and mean curvature

TL;DR

This paper provides a criterion to determine the semisimplicity of indefinite extrinsic symmetric spaces using the shape operator related to the mean curvature vector, enhancing previous results.

Contribution

It improves a prior result by Eschenburg and Kim by establishing a new criterion based on the shape operator for semisimplicity.

Findings

Established a new criterion for semisimplicity

Connected shape operator properties with space semisimplicity

Extended understanding of extrinsic symmetric spaces

Abstract

Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.