Ricci curvature on warped product submanifolds in spheres with geometric applications

Abstract

The goal of this paper is to construct a fundamental theorem for the Ricci curvature inequality via partially minimal isometric warped product immersions into an m-dimensional unit sphere Sm, involving the Laplacian of a well defined warping function, the squared norm of a warping function and the squared norm of the mean curvature. Moreover, the equality cases are discussed in detail and some applications are also derived due to involvement of the warping function. As applications, we provide sufficient condition that the base N1p is isometric to the sphere Sp(λ1p) with constant sectional curvature c=λ1p. The obtained results in the paper give the partial solution of Ricci curvature conjecture, also known as Chen-Ricci inequality obtained by Chen (1999).