Supersymmetric structure of the induced W gravities

We derive the supersymmetric structure present in W-gravities which has bee n already observed in various contexts as Yang-Mills theory, topological fi eld theories, bosonic string and chiral W-3-gravity. This derivation which is made in the geometrical framework of Zucchini, necessitates the introduc tion of an appropriate new basis of variables which replace the canonical f ields and their derivatives. This construction is used, in the W-2-case, to deduce from the Chern-Simons action the Wess-Zumino-Polyakov action.