Riemannian geometry of the space of volume preserving immersions

TL;DR

This paper studies the geometric structure of the space of volume-preserving immersions between manifolds, analyzing its submanifold properties, connections, geodesics, and well-posedness, extending concepts from fluid mechanics to a broader geometric context.

Contribution

It establishes the manifold structure of non-minimal volume-preserving immersions, derives Levi-Civita connections for Sobolev metrics, and proves local well-posedness of geodesic equations.

Findings

Non-minimal volume-preserving immersions form a splitting submanifold.

Derived Levi-Civita connections for natural Sobolev metrics.

Proved local well-posedness of geodesic equations in many cases.

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.