  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.
