  This is a detailed development for the $A_n$ case, of our previous article
entitled "W-Geometries" to be published in Phys. Lett. It is shown that the
$A_n$--W-geometry corresponds to chiral surfaces in $CP^n$. This is comes out
by discussing 1) the extrinsic geometries of chiral surfaces (Frenet-Serret and
Gauss-Codazzi equations) 2) the KP coordinates (W-parametrizations) of the
target-manifold, and their fermionic (tau-function) description, 3) the
intrinsic geometries of the associated chiral surfaces in the Grassmannians,
and the associated higher instanton- numbers of W-surfaces. For regular points,
the Frenet-Serret equations for $CP^n$--W-surfaces are shown to give the
geometrical meaning of the $A_n$-Toda Lax pair, and of the conformally-reduced
WZNW models, and Drinfeld-Sokolov equations. KP coordinates are used to show
that W-transformations may be extended as particular diffeomorphisms of the
target-space. This leads to higher-dimensional generalizations of the WZNW and
DS equations. These are related with the Zakharov- Shabat equations. For
singular points, global Pl\"ucker formulae are derived by combining the
$A_n$-Toda equations with the Gauss-Bonnet theorem written for each of the
associated surfaces.
