Parallel and semiparallel space-like surfaces in pseudo-Euclidean spaces (Q2721682)

Parallel submanifolds in pseudo-Euclidean spaces are characterized locally by the system \(\overline\nabla h=0\). Submanifolds satisfying the integrability condition \(\overline R\circ h=0\) of this system are called semiparallel; geometrically they are second-order envelopes of the parallel submanifolds. Here, \(\overline R\) is the curvature operator of the van der Waerden-Bortolotti connection \(\overline\nabla\) and \(h\) the second fundamental form. 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 not exist totally geodesic minimal semiparallel space-like surfaces.