  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.
