Canonical Reductive Decomposition of Extrinsic Homogeneous Submanifolds

TL;DR

This paper investigates a canonical reductive decomposition for extrinsic homogeneous submanifolds within a homogeneous Riemannian manifold, linking their extrinsic properties with the structure of the ambient space.

Contribution

It introduces a canonical reductive decomposition for orbits of subgroup actions, connecting extrinsic geometry with homogeneous structures and the Ambrose-Singer theorem.

Findings

Established a canonical reductive decomposition for extrinsic homogeneous submanifolds

Linked the decomposition with extrinsic geometric properties

Connected the analysis with the Ambrose-Singer theorem

Abstract

Let $\overline{M}=\overline{G}/\overline{H}$ be a homogeneous Riemannian manifold. Given a Lie subgroup $G\subset \overline{G}$ and a reductive decomposition of the homogeneous structure of $\overline{M}$, we analyze a canonical reductive decomposition for the orbits of the action of $G$. These leaves of the $G$-action are extrinsic homogeneous submanifolds and the analysis of the reductive decomposition of them is related with their extrinsic properties. We connect the study with works in the literature and initiate the relationship with the Ambrose-Singer theorem and homogeneous structures of submanifolds.