Riemannian geometry of the space of volume preserving immersions

Abstract

Given a compact manifold $M$ and a Riemannian manifold $N$ of bounded geometry, we consider the manifold ${\rm Imm} (M,N)$ of immersions from $M$ to $N$ and its subset ${\rm Imm}_\mu (M,N)$ of those immersions with the property that the volume-form of the pull-back metric equals $\mu$. We first show that the non-minimal elements of ${\rm Imm}_\mu (M,N) $ form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics write down the geodesic equation and 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 great importance in fluid mechanics.