  It is shown that, classically, the W-algebras are directly related to the
extrinsic geometry of the embedding of two-dimensional manifolds with chiral
parametrisation (W-surfaces) into higher dimensional K\"ahler manifolds. We
study the local and the global geometries of such embeddings, and connect them
to Toda equations. The additional variables of the related KP hierarchy are
shown to yield a specific coordinate system of the target-manifold, and this
allows us to prove that W-transformations are simply particular diffeomorphisms
of this target space. The W-surfaces are shown to be instantons of the
corresponding non-linear $\sigma$-models.
