Totally geodesic submanifolds of the homogeneous nearly Kähler 6-manifolds and their G2-cones

In this article we classify totally geodesic submanifolds of homogeneous nearly Kähler 6-manifolds, and of the G2-cones over these 6-manifolds. To this end, we develop new techniques for the study of totally geodesic submanifolds of analytic Riemannian manifolds, naturally reductive homogeneous spaces, and Riemannian cones. In particular, we obtain an example of a totally geodesic submanifold with self-intersections in a simply connected homogeneous space.