Chen inequality for general warped product submanifold of Riemannian warped products I×fSm(c)

Abstract

In the present paper, we investigate the geometry and topology of warped product submanifolds in Riemannian warped product I×fSm(c) and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Our main object is to apply geometrically to number structures and obtained applications of Dirichlet eigenvalues problems. Some new results on mean curvature vanishing are presented, and a partial solution can be found to the well-known problem given by S S Chern. The family of Riemannian warped products generalized to Robertson-Walker space-times