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 G is closed under the Lie bracket, then any normal Jacobi operator K of M equals the square of the associated invariant shape operator α. 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 = α2, 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.