On the gradient flows on Finsler manifolds

TL;DR

This paper explores the structure of Finsler manifolds, introduces a curvature functional, and develops a gradient flow approach to evolve metrics towards constant curvature solutions, extending concepts similar to Ricci flow.

Contribution

It provides a novel framework for gradient flows on Finsler manifolds based on the Akbar-Zadeh curvature functional, including analysis of critical points and flow properties.

Findings

Critical points correspond to metrics with constant Ricci-directional curvature.

Gradient flow can evolve metrics towards constant second type scalar curvature.

Comparison with Ricci flow suggests new evolution equations in Finsler geometry.

Abstract

The purpose of this article is to provide a general overview of curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first prove that the space of Finslerian metrics is a Riemannian manifold. Then it is given a decomposition for the tangent space of this Riemannian manifold by means of Riemannian metric and the Berger-Ebin theorem. Next, Finslerian functional is introduced and show that Akbar-Zadeh curvature functional is the example of Finslerian functional. After that, the critical points of Akbar-Zadeh functional are found in two situations. Based on the constant Indicatrix volume and restricting its variations to the point-wise conformal transformations, we prove that the critical points of functional are metrics of constant Ricci-directional curvature. Finally, the gradient flow of Akbar-Zadeh curvature functional in this special direction is introduced as a good candidate for evolving to the metric with constant second type scalar curvature and we compare this new ow with introducing Ricci flow in Finsler geometry.