Parallel and semiparallel space-like surfaces in pseudo-Euclidean spaces

Abstract

Parallel submanifolds in pseudo-Euclidean spaces are characterized locally by the system ∇h = 0. Submanifolds satisfying the integrability condition R • h = 0 of this system are called semiparallel; geometrically they are 2nd-order envelopes of the parallel submanifolds. The existence and geometry of such two-dimensional Riemannian submanifolds (surfaces) are investigated and their complete classification is given. Moreover, it is shown that in E n s with s > 0 do exist not totally geodesic minimal semiparallel space-like surfaces.

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

“…Among others, for semi-parallel submanifolds in real space forms, see [182,183,184,185,186,187,188,189,190,191]; for semi-parallel submanifolds of indefinite space forms, see [192,193]; for semi-parallel submanifolds in Kaehler manifolds, see [194,195,196,197]; for semi-parallel submanifolds in reducible spaces, see [116]; for manifold with semi-parallel geodesic spheres or semi-parallel tubes, see [198]; for semi-parallel submanifolds in contact metric manifolds, see [199,200]; and for semi-parallel submanifolds in other Riemannian manifolds, see [201,202,203]. For some further results on semi-parallel submanifolds, see [2].…”