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M. Camarinha, M. Raffaelli:
"Curvature-adapted submanifolds of bi-invariant Lie groups";
Vortrag: XXIX International Fall Workshop in Geometry and Physics, Universidade Beira Interior, Covilhã, Portugal, online; 07.09.2021 - 10.09.2021.

Given a hypersurface M of a Riemannian manifold Q, one says that M is curvature-adapted (to Q) if, for each p ∈M , the normal Jacobi operator and the shape operator of M commute. The first operator measures the curvature of the ambient manifold along the normal vector of M , whereas the second describes the curvature of M as a submanifold of Q. This condition can be generalized to submanifolds of arbitrary codimension.
In this talk, we will study curvature-adapted submanifolds in a Lie group G equipped with a bi-invariant Riemannian metric. In particular, we shall see that, if the normal bundle of M ⊂ G is abelian [3] (for every p ∈ M , exp(NpM ) is contained in some totally geodesic, flat submanifold of G), then any normal Jacobi operator of M equals the square of the corresponding invariant shape operator.This permits to understand curvature-adaptedness to G in terms of left translations. For example, it turns out that, in the case where M is a hypersurface, the normal Jacobi operator commutes with the ordinary shape operator precisely when the left-invariant extension of each of its eigenspaces remains tangent to M along all the others. As a further consequence of the same result, one observes that any surface in a three-dimensional bi-invariant Lie group is curvature-adapted.