Curvature-adapted submanifolds of semi-Riemannian groups

Abstract

We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group $G$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal Jacobi operator $K$ of $M$ equals the square of the associated invariant shape operator $\alpha$. This permits to understand curvature adaptedness to $G$ geometrically, in terms of left translations. For example, in the case where $M$ is a Riemannian hypersurface, our main result states that the normal Jacobi operator commutes with the ordinary shape operator precisely when the left-invariant extension of each of its eigenspaces has first-order tangency with $M$ along all the others. As a further consequence of the equality $K = \alpha^{2}$, we obtain a new case-independent proof of a well-known fact: every three-dimensional Lie group equipped with a bi-invariant semi-Riemannian metric has constant curvature.