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Description of surfaces associated with \({\mathbb C} P^{N-1}\) sigma models on Minkowski space. (English) Zbl 1092.53044
The authors study two-dimensional smooth and orientable surfaces \(F\) immersed in the Lie-algebra su\((N)\) which are associated with the \({\mathbb C} P^{N-1}\) sigma model defined on two-dimensional Minkowski space. They determine the first and the second fundamental form to \(F\) and compute the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations for \(F.\) Formulae for the Gaussian curvature, the mean curvature and the Willmore functional are computed, too. Then they construct a standard representation of a moving frame of \(F\) immersed in su\((N).\) To this end they use the adjoint representation of the group SU\((N).\) This delivers a normal representation harnessing an orthogonalizing process similar to the method of Gram-Schmidt. The case \(N=2\) leads to surfaces with constant negative Gaussian curvature. An example of this type closes the paper.

53C40 Global submanifolds
53C43 Differential geometric aspects of harmonic maps
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