Submitted Paper
Inserted: 26 apr 2024
Last Updated: 26 apr 2024
Year: 2024
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.
Keywords: Geodesic distance, Hilbert manifold, weak Riemannian metric, infinite dimensional Heisenberg group, sectional curvature
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