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We gave a new very simple proof that the completion of the space of thediffeomorphism of the circle modulo conformal maps with respect to theWeil-Petersson Metric is a complex analytic manifold modeled on the Hilbertspace with 3/2 Sobolev norm. Our proof is based on the analogue of the HadamardTheorem that the exponentila map is a complex analytic map from the tanegntspace of a point of a simply connected manifold to the manifold.

The Hilbert 3/2 Structure and Weil-Petersson Metric on the Space of the Diffeomorphisms of the Circle Modulo Conformal Maps