Two natural metrics and their covariant derivatives on a manifold of embeddings

OnE(M, ⁿ), the Fréchet-manifold of all smooth embeddings of a smooth, compact, closed, orientable manifoldM (of dimensionn-1) into ⁿ two natural metricsG and are considered. The metric plays a central rôle in elasticity theory. Using a generalised notion of the Fréchet derivative their respective spraysS and and the correspoonding Levi-Civita connections are computed. BothS and are smooth in a well defined sense. In contrast toS the spray turns out to be trivial.