On the Michor-Mumford phenomenon in the infinite dimensional Heisenberg group

TL;DR

This paper explores the Michor-Mumford phenomenon in an infinite dimensional Heisenberg group, demonstrating how degenerate distances relate to unbounded sectional curvature, extending previous observations in Fréchet manifolds.

Contribution

It constructs a left invariant weak Riemannian metric in the infinite dimensional Heisenberg group and analyzes its degenerate geodesic and sub-Riemannian distances, linking them to curvature behavior.

Findings

Degenerate geodesic distance is constructed and analyzed.

Sectional curvature can be unbounded and not defined everywhere.

Degeneracy of distance correlates with unbounded curvature on certain planes.

Abstract

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fr\'echet manifolds.