2014/11/30 by Martin Bauer, Martins Bruveris, Peter W. Michor +2 · 1 citation
Mathematics · #math.DG #math.PR #msc:58B20 #msc:58D15
paper · pdf · doi:10.1016/j.difgeo.2016.07.002
published as Differential Geometry and its Applications 49 (December 2016), 23-42 · 22 pages, no figure
arxiv created 2016/03/18 · arxiv updated 2017/08/02
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 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 \rm Immμ(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.