Curvature in sub-Riemannian geometry

Abstract

We study curvature problems on a nearly Riemannian manifold, which is a sub-Riemannian manifold (M, HM, g, VM) whose adapted tensor field given by (2.2) vanishes identically. First, we prove the existence and uniqueness of what we call horizontal Riemannian connection, which is a torsion-free and metric linear connection ∇ on the horizontal distribution HM. Then we define the horizontal sectional curvature of M and study nearly Riemannian space forms, which are nearly Riemannian manifolds of constant horizontal sectional curvature. Several horizontal geometric objects are introduced and studied, for example, horizontal Ricci tensor, horizontal scalar curvature, horizontal Laplacian. Finally, as a prelude to applications to Kaluza–Klein theories we define the horizontal Einstein gravitational tensor field and show that its horizontal divergence vanishes identically on M. Some examples are presented to illustrate the theory developed in the paper.