Sub-Riemannian manifold

A Sub-Riemannian manifold is a triple $(M,\mathscr{H},g_{\mathscr{H}})$ where $M$ is a manifold,$\mathscr{H}$ is a distribution (that is, a linear subbundle of the tangent bundle $TM$ of $M$), and$g_{\mathscr{H}}$ is a metric on $\mathscr{H}$ induced by a fiber inner product on $\mathscr{H}$. Thedistribution $\mathscr{H}$ is often referred to as the horizontal distribution.