  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.
