GEOMETRIC FLOWS ON WARPED PRODUCT MANIFOLD

Abstract

We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold (WPM) by introducing a new notation for the lift vector and the Levi-Civita connection. Using well-established formula, we consider two questions on WPM related to Ricci flow (RF) and hyperbolic geometric flow (HGF). Firstly, we discuss the preserved flow-type problem which says that when the first factor $(M,g)$ and the second factor $(N,h)$ are solutions to the RF (or HGF), the singly WPM $M \times_\lambda N$ is still solution to the RF (or HGF). We obtain some characteristic PDEs satisfied by warping function and also construct some simple examples. Next, we discuss the evolution equations for warping function $\lambda$ and Ricci curvature tensor etc. under RF/HGF. We gain some interesting results, especially adding an assumption with Einstein metric to the second factor.