Geometric Properties and Spectral Estimates on Warped Products

Abstract

We establish an integral inequality for the Ricci curvature of a certain class of warped products $M{\times }_{f}N$, where the equality holds if and only if it is simply a Riemannian product. We also give a sufficient condition for the intersection of a warped product $M=\mathbb{R}{\times }_{f}P$ with a totally geodesic hypersurface $N$ in an arbitrary Riemannian space to be a totally geodesic slice of $M$. In addition, we establish some spectral estimates for the Laplacian of a submanifold $N$ that intersects a warped product in the same ambient manifold.