The Riemannian manifold of all Riemannian metrics

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152773The space of all Riemannian metrics on a smooth second countable finite dimensional manifold is itself a smooth manifold modeled on the space of symmetric (0,2)-tensor fields with compact support. It carries a canonical Riemannian metric which is invariant under the action of the diffeomorphism group. We determine its geodesics, exponential mapping, curvature, and Jacobi fields in a very explicit manner.Differential GeometryFunctional Analysis58D17 58B20The Riemannian manifold of all Riemannian metricstext