Geometry of Submanifolds Derived from $\operatorname{Spin}$-Valued Spectral Problems

We present recent results motivated by Sym's theory of soliton surfaces. Quite general assumptions about the structure of the spectral problem can lead to some specific classes of surfaces. In some cases (including pseudospherical surfaces), this approach is coordinate-independent, which seems a surprising novelty. The Darboux–Bäcklund transformation is formulated in terms of Clifford numbers, which greatly simplifies constructing explicit solutions. Cumbersome computations in matrix representations are replaced with rotations represented by elements of an appropriate $\operatorname{Spin}$ group. Finally, the spectral problem and the spectral parameter are derived purely geometrically in the case of isometric immersions of constant-curvature spaces in spheres and Euclidean spaces.