Geometric Inequalities for Warped Products in Riemannian Manifolds

Mathematics

Bang-Yen Chen¹  AdaraM. Blaga² 

[1] Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA;Department of Mathematics, West University of Timişoara, 300223 Timişoara, Romania;

关键词: warped products;    warped product immersion;    inequality;    space forms;    space of quasi-constant curvature;    eigenfunction;   

DOI  :  10.3390/math9090923

来源: DOAJ

【 摘 要 】

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product $N_1{\times }_f{N}_2$ into any Riemannian manifold $R^m\left(c\right)$ of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature $H^2$ . Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.

【 授权许可】

Unknown