Geometry and Analysis on Nonholonomic Structures on Manifolds
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Structures of nonholonomic nature such as bracket generating distributions and sub-Riemannian structures appear naturally in Control Theory with applications to Robotics (car-like robots), Mechanics (ball bearing structures) and the models of visual perception. Distributions are also basic objects in the geometric theory of differential equations and in geometry of real submanifolds of complex spaces (Cauchy-Riemann geometry). The knowledge of invariants of a geometric structure up to diffeomorphisms of the ambient manifolds or, informally speaking, of the quantities which are independent of local coordinate representations of such structures often gives the essential information about various qualitative properties of the geometric structures. In the last decade we developed the novel variational approach for the unified construction of the differential invariants for a very wide class of geometric structures. This approach takes its origin in Optimal Control Theory and Symplectic Geometry. It extends significantly the set of the nonholonomic structures for which the canonical frames and differential invariants can be constructed explicitly and uniformly compared with all previously existing approaches (including the classical Cartan method of equivalence and the Tanaka theory from 1970). The aim of the project is to use these invariants in order to solve a number of natural problems that were not considered in such generality before. The results of this project will bring new geometric tools for Control Theory and Mathematical Physics by providing efficient and uniform algorithms for computing state-feedback and gauge invariants and explicit geometric optimality conditions for extremals of the corresponding variational problems. We will develop a special MAPLE based software package for computation of our invariants and apply the theory to the qualitative study of control and mechanical systems of practical interest.The main new point of the aforementioned variational approach is that the study of geometry of nonholonomic structures on manifold can be reduced to a simpler (extrinsic) geometry of curves in flag varieties. In terms of these curves we obtain a new discrete basic invariant of the original structure, called the flag symbol and we have an explicit algorithm for construction of the canonical frame for our original structure that depends only on first fixing this discrete information. However, the main properties of these frames and of the invariants they produce are far of being understood. Among the problems that will be addressed in the project are (1) explicit description of the group of symmetries of the most symmetric models with given flag symbol; (2) exploring when the obtained canonical frames are Cartan connections; (3) distinguishing the fundamental set of invariants among all invariants produced by these frames. To reach these goals we will use various tools and techniques from Representation Theory, Algebraic Geometry, and cohomological theory of Lie algebras. Another main theme of the project is comparison theorems in sub-Riemannian geometry. Several essential differences here compared with the Riemannian case form serious obstacles in obtaining sharp analogs of the classical Rauch and Bonnet-Myers comparison theorems. To overcome these obstacles we propose to work on the infinitesimal version of comparison theorems, i.e. given a sub-Riemannian metric to describe directions in the tangent space at this metric to the space of all sub-Riemannian metrics in which consecutive conjugate points along the corresponding extremals become closer. We also shall study natural flows on the space of sub-Riemannian metrics along which the infinitesimal comparison theorem holds.