Totally geodesic hypersurfaces of homogeneous spaces

TL;DR

This paper classifies simply connected homogeneous spaces that contain totally geodesic hypersurfaces, showing they are either products, warped products, or twisted products with explicit warping functions.

Contribution

It provides a complete classification of such spaces and describes the geometric structure and submersion properties associated with these hypersurfaces.

Findings

Spaces are either Riemannian products, warped products, or twisted products.

Totally geodesic hypersurfaces are leaves of homogeneous fibrations.

Case (c) relates to Riemannian submersions onto the universal cover of SL(2).

Abstract

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product of the line and a homogeneous space (with the warping/twisting function given explicitly). In the first case, F is also a Riemannian product; in the last two cases, it is a leaf of a totally geodesic homogeneous fibration. Case (c) can alternatively be characterised by the fact that M admits a Riemannian submersion onto the universal cover of the group SL(2) equipped with a particular left-invariant metric, and F is the preimage of the two-dimensional solvable totally geodesic subgroup.