Totally geodesic submanifolds are perhaps one of the easiest types of submanifolds of Riemannian manifolds one can study, since a maximal totally geodesic submanifold is completely determined by any one of its points and the tangent space at that point. It comes as a bit of a surprise then that classification of such submanifolds — up to an ambient isometry — is a nightmarish and widely open question, even on such a manageable and well-understood class of Riemannian manifolds as symmetric spaces.
We will discuss the theory of totally geodesic submanifolds of symmetric spaces and see that any maximal such submanifold is homogeneous and thus can be completely encoded by some Lie algebraic data called a 'Lie triple'. We will then talk about the duality between symmetric spaces of compact and noncompact type and discover that there is a one-to-one correspondence between totally geodesic submanifolds of a symmetric space and its dual. Finally, we will touch on the known classification in rank one symmetric spaces, namely in spheres and projective/hyperbolic spaces over real normed division algebras. Time permitting, I will demonstrate how all this business comes in handy in other geometric problems on symmetric spaces, e. g. in classification of isometric cohomogeneity one actions.
Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…