Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. (English) Zbl 1083.58010
Authors’ abstract: The \(L^2\)-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type \(M\) in a Riemannian manifold \((N,g)\) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the \(L^2\)-metric.
Reviewer: Hiroaki Shimomura (Kochi)
MSC:
58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
58D15 | Manifolds of mappings |
58D10 | Spaces of embeddings and immersions |
58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |