On the Geometry of W_n gravity

We report work done with T. Jayaraman (hep-th/9405146) in this talk. In a recent paper, Hitchin introduced generalisations of the Teichmuller space of Riemann surfaces. We relate these spaces to the Teichmuller spaces of W-gravity. We show how this provides a covariant description of W-gravity and naturally leads to a Polyakov path integral prescription for W-strings. (Talk presented at the International Colloquium on Modern Quantum Field Theory II at the Tata Institute, Bombay during Jan. 5-11, 1994, to appear in the proceedings)