Riemannian geometry of the space of volume preserving immersions

Given a compact manifold M and a Riemannian manifold N   of bounded geometry, we consider the manifold Imm(M,N)Imm(M,N) of immersions from M to N   and its subset Immμ(M,N)Immμ(M,N) of those immersions with the property that the volume-form of the pull-back metric equals μ  . We first show that the non-minimal elements of Immμ(M,N)Immμ(M,N) form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics, we write down the geodesic equation for which we show local well-posedness in many cases. The question is a natural generalization of the corresponding well-posedness question for the group of volume-preserving diffeomorphisms, which is of importance in fluid mechanics.