Totally geodesic submanifolds of regular Sasakian manifolds (Q411710)

The main result of this paper finds a correspondence between certain types of totally geodesic submanifolds of a homogeneous Sasakian manifold (necessarily regular) and submanifolds of its Hermitian symmetric base space (via the Boothby-Wang fibration). Namely, invariant submanifolds (with respect to the \((1,1)\) tensor of the Sasakian structure) correspond to complex submanifolds of the base, and anti-invariant submanifolds of maximal dimension (necessarily Legendrian with respect to the underlying contact structure) correspond to totally real submanifolds of the base. As a consequence, there are no totally geodesic, contact CR submanifolds in a homogeneous Sasakian manifold. This results extend and complete earlier ones by Yano \& Kon, Leung etc. On the other hand, the result can be seen as a step toward the classification of totally geodesic submanifolds of Sasakian manifolds and as a recipe for producing examples. The proof relies on local computations. The paper contains a very useful up to date presentation of the progress in the problem of classifying totally geodesic submanifolds.