Mathematics

In a connected Banach manifold, equipped with a weak Riemannian metric, we show that the infimum of the lengths among all piecewise smooth curves joining two points (namely, the geodesic "distance") does not define a distance in general. We construct a simple counterxample in l^2. In addition, if a weak Riemannian metric is left invariant on a Banach Lie group, the geodesic "distance" still may not be a distance. Indeed, we construct a counterexample in the infinite dimensional Heisenberg group, and we show how the sectional curvatures can be calculated for some specific planes, confirming a phenomenon first observed by Michor and Mumford. This is a joint work in collaboration with Professor Valentino Magnani (University of Pisa).