Sub-Riemannian homogeneous spaces in dimensions 3 and 4
A sub-Riemannian manifold is a triple \((M,{\mathcal D},g)\), consisting of a manifold \(M\), a distribution \({\mathcal D}\) on \(M\), and a metric \(g\) on \({\mathcal D}\). In this paper, the dimension of \(M\) is 3 or 4, \({\mathcal D}\) has codimension 1 and is contact or odd-contact, that is, the Levi-Form on \({\mathcal D}\) (which for vector fields with values in \({\mathcal D}\) is the Lie-bracket mod \({\mathcal D})\) is nondegenerate \((\dim M=3)\) or has one-dimensional kernel \((\dim M=4)\). In either case there exists a unique, not torsion-free connection, fitting the data of the sub-Riemannian manifold in a specific manner. In the 3-dimensional case there exist 4 scalar invariants (one of them is the sectional curvature of \({\mathcal D})\), which, when being constant, characterize local isometry of sub-Riemannian manifolds. A sub-Riemannian manifold is called homogeneous, if it admits a transitive Lie group of sub-Riemannian isometries. The homogeneous sub-Riemannian manifolds are classified, using the classification of sub-Riemannian symmetric spaces of \textit{R. S. Strichartz} [J. Differ. Geom. 24, 221-263 (1986; Zbl 0609.53021)]. There are two new classes of sub-Riemannian manifolds which are homogeneous but not symmetric. The relation between CR-manifolds and sub-Riemannian manifolds is used to construct all simply connected, homogeneous, nondegenerate 3-dimensional CR-manifolds. For \(\dim M=4\), the simply connected sub-Riemannian symmetric manifolds are in 1-1 correspondence with an object called a sub-orthogonal involutive Lie algebra. With the help of this correspondence, the 4-dimensional sub-Riemannian symmetric manifolds are classified.
scientific article; zbMATH DE number 929916