A PROPOSAL FOR THE GEOMETRY OF W-N-GRAVITY

We relate the Teichmuller spaces obtained by Hitchin to the Teichmulle r spaces of WA(n)-gravity. The relationship of this space to W-gravity is obtained by identifying the flat PSL(n + 1,R) connections of Hitch in to generalised vielbeins and connections. This is explicitly demons trated for WA(2) = W3 gravity, We show how W-diffeomorphisms are obtai ned in this formulation. We find that particular combinations of the g eneralised connection play the role of projective connections. We thus obtain W-diffeomorphisms in a geometric fashion without invoking the presence of matter fields. This description in terms of vielbeins natu rally provides the measure for the gravity sector in the Polyakov path integral for W-strings.