  I show that the generalized Beltrami differentials and projective connections
which appear naturally in induced light cone $W_n$ gravity are geometrical
fields parametrizing in one-to-one fashion generalized projective structures on
a fixed base Riemann surface. I also show that $W_n$ symmetries are nothing but
gauge transformations of the flat ${SL}(n,{\bf C})$ vector bundles canonically
associated to the generalized projective structures. This provides an original
formulation of classical light cone $W_n$ geometry. From the knowledge of the
symmetries, the full BRS algebra is derived. Inspired by the results of recent
literature, I argue that quantum $W_n$ gravity may be formulated as an induced
gauge theory of generalized projective connections. This leads to projective
field theory. The possible anomalies arising at the quantum level are analyzed
by solving Wess-Zumino consistency conditions. The implications for induced
covariant $W_n$ gravity are briefly discussed. The results presented, valid for
arbitrary $n$, reproduce those obtained for $n=2,3$ by different methods.

High Energy Physics - Theory