Measures of transverse paths in sub-Riemannian geometry
Let \(M\) be a \(\mathcal{C}^{\infty}\) manifold, let \(\mathcal{D} \subset TM\) be a distribution on \(M\) and let \(g\) be a \(\mathcal{C}^{\infty}\) Riemannian metric on \(\mathcal{D}\), so that \((M, \mathcal{D}, g)\) is a sub-Riemannian manifold. The authors assume that Chow's condition is satisfied, namely if \(\mathcal{D}^{s}\) denotes the linear span of all brackets of degree \(\leq s\) of the vector fields tangent to \(\mathcal{D}^{1} = \mathcal{D}\), then at every point \(p \in M\) there exists an integer \(r = r(p)\) such that \(\mathcal{D}^{r} (p) = T_{p} M\). The horizontal paths are those paths which are tangent to \(\mathcal{D}\). Their lengths are obtained as in the Riemannian geometry by integrating the norm of their corresponding tangent vectors. The Chow condition implies that one can join any two points of the manifold by a horizontal path, and therefore a distance \(d\) can be defined. A transverse path is one whose tangent vector does not belong to the distribution. In the paper under review the authors introduce a length or a measure for a transverse path. Their definition is based on a careful study of the metric structure of the tangent space. The metric tangent space, called here tangent cone is itself a sub-Riemannian manifold and has moreover an algebraic structure, namely it is a group at a regular point. The authors propose an intrinsic definition of the tangent cone at any point. Furthermore, they introduce a filtration of the tangent cone and relate it to the natural filtration of the tangent space defined by the filtration. A class of lengths of paths is defined by integration of infinitesimal measures on the tangent cone. One of those lengths coincides with the usual length. As an application, the measures for contact and Martinet sub-Riemannian manifolds are computed.
infinitesimal measure on the tangent cone
scientific article; zbMATH DE number 2103496
scientific article