The Michor-Mumford conjecture in Hilbertian H-type groups

Abstract

We introduce infinite dimensional Hilbertian H-type groups equipped with weak, graded, left invariant Riemannian metrics. For these Lie groups, we show that the vanishing of the geodesic distance and the local unboundedness of the sectional curvature coexist. The result validates a deep phenomenon conjectured in an influential 2005 paper by Michor and Mumford, namely, the vanishing of the geodesic distance is linked to the local unboundedness of the sectional curvature. We prove that degenerate geodesic distances appear for a large class of weak, left invariant Riemannian metrics. Their vanishing is rather surprisingly related to the infinite dimensional sub-Riemannian structure of Hilbertian H-type groups. The same class of weak Riemannian metrics yields the nonexistence of the Levi-Civita covariant derivative.

arXiv comment18 pages

References in corpus (6)

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line
• Vanishing distance phenomena and the geometric approach to SQG
• Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds
• An introduction to infinite-dimensional differential geometry
• Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves