Rigid paths of generic 2-distributions on 3-manifolds (Q1902026)

Let \(M\) be a smooth connected manifold and \(E\) be a nonholonomic \(k\)-distribution on \(M\) (a smooth \(k\)-dim subbundle of the tangent bundle to \(M)\). For points \(a,b\) in \(M\) the abnormal (rigid) paths from \(a\) to \(b\) are the singular (resp. the isolated) points of the space, with the natural \(C^1\)-topology, of smooth paths \(\gamma\) from \(a\) to \(b\) with \(\dot\gamma \in E\). This paper gives a complete and explicit description of all rigid paths of generic 2-distributions on 3-manifolds and also includes a complete description of abnormal and (1) locally rigid paths of Martinet distributions, and of 2-distributions on 5-manifolds with growth vector and (2) rigid paths of Engel distributions. Also dealt with are local existence theorems for rigid paths of generic 2-distributions on a manifold of dimension \(\geq 5\) and the absence of abnormal paths for strong bracket-generating distributions, and of rigid paths passing through generic \(k\)-distributions on \((n\leq k+k(k-1)/2)\)-dimensional manifolds.