Riemannian Exponential Maps of the Diffeomorphism Group of T2

We study the exponential maps induced by right-invariant weak Riemannian metrics of Sobolev type of order k 0 on the Lie group of smooth, orientation preserving dieomorphisms of the two dimensional torus. We prove that for k 1, but not for k = 0, each of them defines a smooth Frechet chart of the identity.