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For M a closed manifold or the Euclidean space R n we present a detailed proof of regularity properties of the composition of H s -regular diffeomorphisms of M for s > 1 2 dim M + 1.

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics µ (k) (k ≥ 0) on the Virasoro group Vir and show that for k ≥ 2, but not for k = 0, 1, each of them defines a smooth Fréchet chart of the unital element e ∈ Vir. In particular, the geodesic exponential map corresponding to the KdV equation (k = 0) is not a local diffeomorphism near the origin.

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