Microflexiblity and local integrability of horizontal curves

Let 𝜉 be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of control theory) has infinite codimension within the space of germs of smooth curves tangent to 𝜉. We formalize this as an asymptotic statement about finite jets of tangent curves. This solves, in the analytic setting, a conjecture of Eliashberg and Mishachev regarding an earlier claim by Gromov about the microflexibility of the tangency condition.

From these statements it follows, by an argument due to Gromov, that the ℎ-principle holds for maps and immersions transverse to 𝜉.