Curvature in the group of measure-preserving diffeomorphisms of the Klein bottle

Speaker: René Langøen, PhD. student @ the University of Bergen        

Abstract: In this talk I introduce the diffeomorphism group of a manifold, with special focus on the diffeomorphism groups of the torus and the Klein bottle. The Lie algebra of a diffeomorphism group of a manifold is given by the vector fields on the manifold.  The torus is a double orientation cover of the Klein bottle implying a direct relation between vector fields on the torus and the Klein bottle. We use this relation to calculate curvature in the diffeomorphism group of the Klein bottle, in particular, we calculate sectional curvature and an infinite dimensional version of the Ricci curvature. The talk is based on recent work with Boris Khesin (University of Toronto) and Irina Markina (UiB).