  The objective of this paper is to construct and investigate smooth orientable
surfaces in $R^{N^2-1}$ by analytical methods. The structural equations of
surfaces in connection with $CP^{N-1}$ sigma models on Minkowski space are
studied in detail. This is carried out using moving frames adapted to surfaces
immersed in the $su(N)$ algebra. The first and second fundamental forms of this
surface as well as the relations between them as expressed in the
Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The Gaussian
curvature, the mean curvature vector and the Willmore functional expressed in
terms of a solution of $CP^{N-1}$ sigma model are obtained. An example of a
surface associated with the $CP^1$ model is included as an illustration of the
theoretical results.
