On local geometry of non-holonomic rank 2 distributions

Abstract

Abstract. In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in R 5 . We solve the analogous problem for germs of generic rank 2 distributions in R n for n > 5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold WD of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n−1)-dimensional fiber bundle over WD with the structure group of all Möbius transformations, preserving 0. The paper is the detailed exposition of the constructions and the results, announced in the short note [8].

Duality of Singular Paths for (2, 3, 5)-Distributions

“…It is shown also that similar double fibrations exist for rank 2 distributions on manifolds of dimension n ≥ 5 and also for distributions of higher rank in [8] [9] implicitly and in [10] explicitly.…”

“…Note that, moreover in [8] [9], it is shown that cone structures exist for rank 2 distributions on n-dimensional manifolds for arbitrary n ≥ 5, and the cone structures were extensively used for constructions of the structures of absolute parallelism and invariants of such distributions. It is shown also that similar double fibrations exist for rank 2 distributions on manifolds of dimension n ≥ 5 and also for distributions of higher rank in [8] [9] implicitly and in [10] explicitly.…”

Twistor Space for Rolling Bodies

“…On the other extreme, a rank two distribution is called generic, or (2,3,5), as e.g. in [6,7], if we have:…”

“…They look ugly, and they all involve the derivatives of the function ρ up to the sixth order. We treated these equations as algebraic equations for ρ (6) , ρ (5) and ρ (4) , and used the same trick as in the proof of Theorem 4.2. Namely, we algebraically solved equation for ρ (4) , differentiated it, and compared it with the ρ (5) obtained algebraically.…”

“…Namely, we algebraically solved equation for ρ (4) , differentiated it, and compared it with the ρ (5) obtained algebraically. Then we did the same for ρ (5) and ρ (6) . This produced a unique compatibility condition, obviously of the third order in the derivatives of ρ, which reads:…”