Canonical Reductive Decomposition of Extrinsic Homogeneous Submanifolds

Let $\bar{M}=\bar{G}\mathrm{/}\bar{H}$ be a homogeneous Riemannian manifold. Given a Lie subgroup $G\subset \bar{G}$ and a reductive decomposition of the homogeneous structure of $\bar{M}$, 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.