A classification of sub-Riemannian manifolds
It is well known that Riemannian surfaces can be classified according to the properties of the collection of harmonic functions defined on them. In [\textit{I. Holopainen}, Ann. Acad. Sci. Fenn., Ser. A I, Diss. 74 (1990; Zbl 0698.31010)] and [\textit{I. Holopainen \text{and} S. Rickman}, Potential Anal.\ 2, No. 1, 37-66 (1993; Zbl 0771.53019)], the classification theory of Riemannian surfaces were generalized to the case of Riemannian manifolds. The latter were classified by the properties of solutions to degenerate elliptic equations similar to the \(p\)-Laplacian. In the article under review, the authors obtain an analogous classification of sub-Riemannian manifolds. It is based on the nonlinear potential theory of degenerate subelliptic equations. Some applications of this classification theory to the quasiregular mappings on Carnot groups are considered.
scientific article; zbMATH DE number 1432316