In finite-dimensional geometry the Riemannian metric is an isomorphism between the tangent and the cotangent bundle. In infinite dimensions the metric is always injective but may fail to be surjective. Accordingly, one distinguishes between two classes of Riemannian metrics: weak and strong ones. In this talk I will discuss several differences between weak and strong Riemannian metrics in terms of existence of the geodesic equation, properties of the geodesic distance, and the theorem of Hopf-Rinow. I will present both the the general theory and some specific examples, including l^2 spaces and diffeomorphism groups, in order to demonstrate the large variety (and pitfalls) of this infinite dimensional setting.
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487