Geometric Properties and Spectral Estimates on Warped Products

TL;DR

This paper investigates geometric inequalities and spectral estimates for warped product manifolds, providing conditions for when they are Riemannian products and analyzing their intersections with geodesic hypersurfaces.

Contribution

It introduces new integral inequalities for Ricci curvature on warped products and offers spectral bounds for Laplacians on intersecting submanifolds.

Findings

Equality in Ricci curvature inequality characterizes Riemannian products.

Provides a sufficient condition for intersections with totally geodesic hypersurfaces to be totally geodesic slices.

Establishes spectral estimates for the Laplacian on submanifolds intersecting warped products.

Abstract

We establish an integral inequality for the Ricci curvature of a certain class of warped products $M\times_fN$, 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 $M=\mathbb{R}\times_fP$ 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.