The Michor–Mumford Conjecture in Hilbertian H-Type Groups

In an influential 2005 paper, Michor and Mumford conjectured that in an infinite dimensional weak Riemannian manifold the vanishing of the geodesic distance is linked to the local unboundedness of the sectional curvature. We introduce infinite dimensional Hilbertian H-type groups equipped with any weak, graded, left invariant Riemannian metric. For these Lie groups we verify the above conjecture by showing that the vanishing of the geodesic distance and the local unboundedness of the sectional curvature coexist. We also observe that our class of weak Riemannian metrics yields the nonexistence of the Levi-Civita covariant derivative.