The Ricci flow as a geodesic on the manifold of Riemannian metrics

Finslerian geodesics on Frechet manifolds

“…Suppose there exists such a covariant derivative. If for all X, Y and Z we compute (10), then by subtracting the sum of the first two from the last one and applying the torsion-free property of a covariant derivative we obtain…”

“…Let p ∇ n be the other covariant derivatives satisfying (10). The right-hand side of ( 11) does not depend on the covariant derivatives, therefore, for all n P N we have…”

“…So, the Euler-Lagrange equations do not hold, therefore, gptq is not geodesic. This result is proved in [9] by using the geodesic equation on the manifold of Riemannian metrics which is considered as the projective limit of Banach manifolds.…”

“…Another recent approach to study geodesics on Fréchet manifolds is by considering these manifolds as projective limits of Banach manifolds, cf. [9,10].…”

A Simple Proof of the Short-time Existence and Uniqueness for Ricci Flow

“…One way is a projective limit approach where we consider a special type of Fréchet manifolds, including the space of Riemannian metrics, which are obtained as projective limit of Banach manifolds (see [3]). Using this method a simpler proof for the short-time existence for Ricci flow was established in [8]. But this approach also has a difficulty, one needs to establish the existence of projective limits of Banach corresponding factors of geometrical and topological objects which would not be always easy.…”