Submanifolds with parallel mean curvature vector in Riemannian and indefinite space forms

Abstract

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we survey some classical and recent results on submanifolds with parallel mean curvature vector. Special attention is paid to the classification of space-like and Lorentz surfaces with parallel mean curvature vector in Riemannian and indefinite space forms.

46 pages

Cited by in corpus (8)

• Quasi-minimal Lorentz Surfaces with Pointwise 1-type Gauss Map in Pseudo-Euclidean 4-Space
• Some open problems and conjectures on submanifolds of finite type: recent development
• Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes
• Meridian Surfaces with Parallel Normalized Mean Curvature Vector Field in Pseudo-Euclidean 4-space with Neutral Metric
• Uniformly Area Expanding Flows in Spacetimes
• Meridian Surfaces on Rotational Hypersurfaces with Lightlike Axis in ${\mathbb{E}}_2^4$
• PMC Biconservative Surfaces in Complex Space Forms
• A comprehensive survey on parallel submanifolds in Riemannian and pseudo-Riemannian manifolds