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

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824-1027, USA
Academic Editor: Sorin Dragomir
Axioms 2019, 8(4), 120; https://doi.org/10.3390/axioms8040120
Received: 17 October 2019 / Revised: 27 October 2019 / Accepted: 28 October 2019 / Published: 30 October 2019
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
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.
Keywords: parallel submanifold; real space form; complex space form; totally real submanifolds; Kaehler submanifolds; light cone; Thurston 3D geometries; Bianchi–Cartan–Vranceasu spaces parallel submanifold; real space form; complex space form; totally real submanifolds; Kaehler submanifolds; light cone; Thurston 3D geometries; Bianchi–Cartan–Vranceasu spaces
MDPI and ACS Style

Chen, B.-Y. A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds. Axioms 2019, 8, 120.

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