Geometric Inequalities for Warped Products in Riemannian Manifolds

Abstract

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 N1×fN2 into any Riemannian manifold Rm(c) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H2 . 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.

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

“…Remark 10. For minimal Lagrangian submanifolds of a complex space form, inequality (32) of Theorem 18 and the improved inequalities (33) of Theorem 20 and (36) of Theorem 22 are the same.…”

“…, n k ) ∈ S(n), there exists a non-minimal Lagrangian submanifold satisfying the equality case of (33). This theorem implies that the inequality (33) in Theorem 20 can not be improved further. Remark 9.…”

“…For instance, it yields the following. For further results in this direction, see [31][32][33][34][35][36] among many others.…”

“…Lagrangian submanifolds in complex space forms satisfying the improved inequalities (33) for δ(2) have been studied extensively in [7,[52][53][54][57][58][59][60][61][62][63] among others. For δ(2, 2)-ideal Lagrangian submanifolds, we refer to [16,18,64].…”

“…Remark 11. After these results, there are many research articles published by various authors for different types of warped product submanifolds in different ambient manifolds (see, e.g., [33,82,83] and recent survey articles published in the two volumes [84,85]).…”