Finslerian geodesics on Fréchet manifolds

Abstract

We establish a framework, namely, nuclear bounded Fréchet 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.

References in corpus (3)

• Geometry of Bounded Frechet Manifolds
• On the generalization of the Darboux theorem
• A simple proof of the short-time existence and uniqueness for Ricci flow

Cited by in corpus (1)

• An introduction to infinite-dimensional differential geometry