On Carnot-Caratheodory metrics.

*(English)*Zbl 0554.53023In this paper the author begins an investigation of the local geometry of certain degenerate metrics on manifolds. If a Riemannian manifold has a distribution of k-planes satisfying a strong non-involutivity condition (Hörmander’s condition), one may define the distance between two points as the infimum of the lengths of all horizontal (i.e., tangent a.e. to the distribution) curves joining the two points. Such metrics arise naturally in a variety of situations, including ideal boundaries of certain symmetric spaces, the study of hypo-elliptic operators, and in control theory.

The author proves that, generically, the tangent cone of such a metric space at a point has a canonical degenerate metric structure associate to a nilpotent Lie group. He also finds a formula for the Hausdorff dimension of such a metric in terms of the infinitesimal parameters of the distribution (this turns out to be the homogeneous dimension of the associated nilpotent Lie group). Both theorems are based on work of Rothschild-Stein concerning hypoelliptic operators.

The author proves that, generically, the tangent cone of such a metric space at a point has a canonical degenerate metric structure associate to a nilpotent Lie group. He also finds a formula for the Hausdorff dimension of such a metric in terms of the infinitesimal parameters of the distribution (this turns out to be the homogeneous dimension of the associated nilpotent Lie group). Both theorems are based on work of Rothschild-Stein concerning hypoelliptic operators.