Geometric flows on warped product manifold

TL;DR

This paper derives a unified formula for Ricci curvature on warped product manifolds, facilitating the study of Ricci flow and hyperbolic geometric flow, and analyzes the evolution of warping functions and metrics.

Contribution

It introduces a new notation and unified formula for Ricci curvature on warped product manifolds, enabling analysis of geometric flows and evolution equations.

Findings

Unified Ricci curvature formula for warped products

Characterization of warping function behavior under RF and HGF

Evolution equations for metrics and Ricci curvature

Abstract

We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and hyperbolic geometric flow (HGF) and evolution equations on warped product manifold. We characterize the behavior of warping function under RF and under HGF. Simultaneously, we give some simple examples to illustrate the existence of such warping function solution. In addition, we also gain the evolution equations for metrics and Ricci curvature on a general warped product manifold and specific warped product manifold whose second factor manifold is of Einstein metric.