Finslerian geodesics on Fr\'{e}chet manifolds

TL;DR

This paper develops a framework for studying geodesics on infinite-dimensional Fréchet manifolds with Finsler structures, establishing existence, minimality, and characterizing geodesics via Euler-Lagrange equations, with applications to Ricci flow.

Contribution

It introduces a novel approach to analyze geodesics on nuclear bounded Fréchet manifolds with Finsler structures, extending geometric analysis to infinite dimensions.

Findings

Existence of local geodesics on these manifolds.

Geodesics are length minimizing in a local sense.

Ricci flow solutions on Einstein manifolds are not geodesic.

Abstract

We establish a framework, namely, nuclear bounded Fr\'{e}chet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.