The Michor-Mumford conjecture in Hilbertian H-type groups

TL;DR

The paper addresses the Michor–Mumford conjecture in infinite dimensional Hilbertian H-type groups endowed with strictly weak, graded left-invariant metrics. It develops a framework of weak Finsler, sub-Finsler, and Riemannian geometries on these groups and analyzes geodesic distances and curvature. The authors prove that degenerate geodesic distances occur alongside local unbounded curvature, establish nonexistence of Levi-Civita covariant derivatives for strictly weak metrics, and derive explicit curvature blow-up via Arnold's formula, thereby validating the Michor–Mumford phenomenon in this setting. The results illuminate the interaction between sub-Riemannian structure and curvature in infinite dimensions and suggest directions for broader generalizations.

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.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2: Characterization of points with vanishing distance
• Theorem 1.3
• Theorem 1.4: Unboundedness of the sectional curvature
• Remark 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Example 3.1
• Remark 3.2