Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

By Peter W. Michor, David Mumford. Documenta Mathematica, 2004

232 citations recorded by the source corpus.

The L 2 -metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian man- ifold (N;g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic dis- tance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all difieomorphism groups for the L 2 -metric.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a985a2a9ddb30dc27fe3e64fc9cafa5bbeec37c

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In Applied Mathematics

1. B. Clarke
2. Martins Bruveris
3. Peter W. Michor
4. Martin Bauer

In Geometry and Topology

1. B. Clarke
2. Martins Bruveris
3. Philipp Harms
4. Martin Bauer
5. Peter W. Michor

In Mathematical Physics

1. B. Clarke
2. Martins Bruveris
3. Peter W. Michor
4. Martin Bauer
5. Tom Leinster
6. Michel Waldschmidt

Citations in the corpus

• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group (2011, background): “Here Jn/2−1 denotes the Bessel function of the first kind, which is given by Jα(r) = 1 π ∫ π 0 cos(αt− r sin t) dt− sin(απ) π ∫ ∞ 0 e−r sinh(t)−αt dt , and Ks−n2 denotes the modified Bessel function of second kind, which is given by Kν(r) = Γ(ν + 12 )(2r) ν √ π ∫ ∞ 0 cos t (t2 + r2)ν+ 1 2 dt .”
• The metric geometry of the manifold of Riemannian metrics over a closed manifold (2009, methodology): “mensional Riemannian manifolds, we can use γ to define a distance between points of N by taking the infima of lengths of paths. It is then clear that this distance function is a pseudometric, but as in [11, 12], it may fail to be positive definite. The problem in showing positive-definiteness on a weak ”
• Some Geometric Evolution Equations Arising as Geodesic Equations on Groups of Diffeomorphisms Including the Hamiltonian Approach (2006, background): “This section is based on [40].”
• Convergence of the thresholding scheme for multi-phase mean-curvature flow (2016, unknown): “[26].”