Ricci–Yamabe maps for riemannian flows and their volume variation and volume entropy

Abstract

The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow g(t) . The considered flow in covariant symmetric 2-tensor fields will be called Ricci–Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of g(t) . Due to the signs of considered scalars the Ricci–Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most commonly addressed situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane.