Description of surfaces associated with $C{P}^{N-1}$ sigma models on Minkowski space

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 $C{P}^{N-1}$ 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 $C{P}^{N-1}$ sigma model are obtained. An example of a surface associated with the $C{P}^1$ model is included as an illustration of the theoretical results.