CarnotCarathéodoryMetrik

The Carnot-Carathéodory metric, also known as the sub-Riemannian metric or Hörmander metric, is a concept in differential geometry that generalizes the standard Riemannian metric. It is defined on a manifold equipped with a distribution, which is a subbundle of the tangent bundle. Instead of considering curves whose tangent vectors lie in the entire tangent space, the Carnot-Carathéodory metric restricts tangent vectors to lie within the specified distribution.

The distance between two points in this metric is defined as the infimum of the lengths of absolutely continuous curves connecting the points, whose tangent vectors at every point belong to the distribution. These curves are often referred to as admissible curves. The length is calculated using a quadratic form defined on the distribution, similar to how lengths are computed in Riemannian geometry.

A key property of the Carnot-Carathéodory metric is that it satisfies the Carathéodory-Carathéodory condition, which ensures that the metric is well-defined and that paths can connect any two points, provided the distribution has sufficient rank. This metric is particularly important in the study of controllability and reachability in mechanical systems and control theory. It also finds applications in areas such as geometric measure theory and sub-Riemannian geometry. The metric is named after Léonard Carnot, Julius Plücker, and Constantin Carathéodory, reflecting its origins and development.