Carnot-Caratheodory spaces (or sub-Riemannian spaces) appear in many theoretical and applied areas such as physics, economics, subelliptic equations, neurobiology, robotechnics, astrodynamics etc. In particular, it can be considered as an interpretation of geometric control theory which is an ideal tool for the application of "pure" mathematics results to some real-life problems.
In our research, we accent on Carnot-Caratheodory spaces with minimal smoothness assumptions since it is unknown whether a modelling space for some certain applied problem possesses any "regularity" properties or not. One of the main results is a comparison estimate for the difference between an initial space and a local Carnot group. Studying such differences is important due to the fact that nilpotent approximations of sub-Riemannian problems are often more convenient to solve than the initial problems on an arbitrary Carnot-Caratheodory space. Our estimate is O(ε1+α/M) if basis vector fields belong to the class C1,α. This result is new even for the smooth vector fields, and it implies a variety of properties including "classical" theorems known before only for the "smooth" case.
These theorems are applied to a solution of complicated open problems in sub-Riemannian geometry such as the proof of sub-Riemannian area and coarea formulas, and description of minimality criteria for graphs on Carnot groups. Proofs required the creation of essentially new methods (new even for the classical Euclidean space) and discovery of new fine properties of sub-Riemannian structures. The formulas are derived in the explicit form convenient for various applications.
The research is supported by the Grant of the Government of Russian Federation for the State Support of Researches (Agreement No 14.B25.31.0029).