On the gradient flows on Finsler manifolds

Abstract

The purpose of this paper is to provide a general overview of a curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first make some differentiable structures on a domain of Finslerian functionals. Then by means of the global inner product and the Berger–Ebin Theorem, we make some decomposition for the tangent space of the manifold of all Finslerian metrics. Next, we study Akbar–Zadeh curvature functional as a Finslerian functional and we find the critical points of this functional in the pointwise conformal metric direction. Through this way, we introduce a gradient flow on compact Finsler manifold. Finally, we compare this new flow with introducing Ricci flow in Finsler geometry.