Submanifolds of indefinite quaternion Kähler manifolds

Abstract

In this chapter, we first recall the structure of indefinite quaternion Käahler manifolds. Then, we give a review of Riemannian submanifolds of quaternion Käahler manifolds. We study the geometry of real lightlike hypersurfaces, the structure of lightlike submanifolds, both, of indefinite quaternion Kähler manifolds and show that a quaternion lightlike submanifold is always totally geodesic. This result implies that the study of lightlike submanifolds, other than quaternion lightlike submanifolds, is interesting. Then, we deal with the geometry of screen real submanifolds in detail. As a generalization of real lightlike hypersurfaces of quaternion Kähler manifolds, we introduce QR-lightlike submanifolds. We show that the class of QR-lightlike submanifolds does not include quaternion lightlike submanifolds and screen real submanifolds. Then, we introduce and study the geometry of screen QR-lightlike and screen CR-lightlike submanifolds as generalizations of quaternion lightlike submanifolds and screen real submanifolds, and provide examples for each class of lightlike submanifolds of indefinite quaternion Kähler manifolds.