Surfaces associated with theta function solutions of the periodic 2D-Toda lattice

The objective of this paper is to present some geometric aspects of surfaces associated with theta function solutions of the periodic 2D-Toda lattice. For this purpose we identify the $(N^2-1)$-dimensional Euclidean space with the ${\frak su}(N)$ algebra which allows us to construct the generalized Weierstrass formula for immersion for such surfaces. The elements characterizing surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations, the Gaussian curvature, the mean curvature vector and the Wilmore functional of a surface are expressed explicitly in terms of any theta function solution of the Toda lattice model. We have shown that these surfaces are all mapped into subsets of a hypersphere in $\mathbb{R}^{N^2-1}$. A detailed implementations of the obtained results are presented for surfaces immersed in the ${\frak su}(2)$ algebra and we show that different Toda lattice data correspond to different subsets of a sphere in $\mathbb{R}^3$.