Totally geodesic orbits of isometries

TL;DR

This paper investigates conditions under which singular orbits in cohomogeneity one Riemannian manifolds are totally geodesic and classifies certain positively curved manifolds with specific symmetry properties.

Contribution

It provides simple criteria for identifying totally geodesic singular orbits and classifies compact positively curved manifolds with non-semisimple symmetry groups and hypersurface orbits.

Findings

Established criteria for totally geodesic singular orbits.

Classified certain positively curved manifolds with specific symmetry groups.

Identified conditions for hypersurface orbits in these manifolds.

Abstract

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non semisimple Lie group with an hypersurface orbit.