Sub-Riemannian manifold

Summary

A sub-Riemannian manifold is a smooth manifold where motion is allowed only along chosen horizontal directions. These form a subbundle of the tangent bundle, and they carry an inner product. Distance is the infimum of lengths of curves that are tangent to the horizontal directions at every time. Assume that the horizontal directions and their Lie brackets span all directions. Then, by the Chow–Rashevskii theorem, any two points of a connected manifold are joined by a horizontal curve, and the distance is finite. Sub-Riemannian geometry appears in control theory, in the study of hypoelliptic operators and in the geometry of the Heisenberg group.