A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper, we provide a comprehensive survey on this important class of submanifolds.

A survey on sufficient conditions for geodesic completeness of nondegenerate hypersurfaces in Lorentzian geometry

This is a survey of the principal results about the geodesic completeness of nondegenerate hypersurfaces in Lorentzian manifolds from a structural point of view. Some of these results retain their validity in the case of semi-Riemannian submanifolds in semi-Euclidean spaces, as well.