  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
$\mathbb{R}^3$.
