Rolling balls and Octonions

In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain geometric meaning of basic invariants of vector distributions

Well-posed infinite horizon variational problems on a compact manifold

We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i. e. a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis we construct a well-projected to M invariant Lagrange submanifold of the extremals' flow in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics

On the Hausdorff volume in sub-Riemannian geometry

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4 on every smooth curve) but in general not C^5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.Comment: Accepted on Calculus and Variations and PD

Nurowski's conformal structures for (2,5)-distributions via dynamics of abnormal extremals

As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic vector distribution on a 5-dimensional manifold M one can assign the canonical conformal structure of signature (3,2). His construction is based on the properties of the special 12-dimensional coframe bundle over M, which was distinguished by E. Cartan during his famous construction of the canonical coframe for this type of distributions on some 14-dimensional principal bundle over M. The natural question is how "to see" the Nurowski conformal structure of a (2,5)-distribution purely geometrically without the preliminary construction of the canonical frame. We give rather simple answer to this question, using the notion of abnormal extremals of (2,5)-distributions and the classical notion of the osculating quadric for curves in the projective plane. Our method is a particular case of a general procedure for construction of algebra-geometric structures for a wide class of distributions, which will be described elsewhere. We also relate the fundamental invariant of (2,5)-distribution, the Cartan covariant binary biquadratic form, to the classical Wilczynski invariant of curves in the projective plane.Comment: 13 page