GEOMETRY OF HIGGS AND TODA FIELDS ON RIEMANN SURFACES

We discuss geometrical aspects of Higgs systems and Toda field theory in the framework of the theory of vector bundles on Riemann surfaces o f genus greater than one. We point out how Toda fields can be consider ed as equivalent to Higgs systems - a connection on a vector bundle E together with an End(E)-valued one form both in the standard and in th e Conformal Affine case. We discuss how variations of Hedge structures can arise in such a framework and determine holomorphic embeddings of Riemann surfaces into locally homogeneous spaces, thus giving hints t o possible realizations of W-n-geometries.