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

Abstract

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.

On Some Weingarten Surfaces in the Special Linear Group SL(2,R)

“…For the sake of simplicity, we kept the same notation. By using (25), it is straightforward to show that (∂ x , ∂ y ) λ = (1−λ) sin 2θ cos 2θ 4y 2…”

“…It is known by Patragenaru (see [24]) that all left-invariant metrics on SU(1, 1) are isometric to one of the 3-parameter families of metrics g(c 1 , c 2 , c 3 ) with c 1 ≥ c 2 > 0 > c 3 , and its isometry group has dimension 4 if and only if c 1 = c 2 . This family of metrics is obtained as follows: (see, e.g., [5,25]):…”