  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.
