Microflexiblity and local integrability of horizontal curves

TL;DR

This paper proves that in the analytic setting, the space of singular tangent curves is negligible compared to all smooth tangent curves, confirming a conjecture about microflexibility and the applicability of the h-principle for certain geometric conditions.

Contribution

It formalizes and proves the microflexibility of tangent conditions for analytic bracket-generating distributions, resolving a conjecture and extending the h-principle to new geometric contexts.

Findings

Singular tangent germs have infinite codimension within smooth tangent germs.

The h-principle applies to maps and immersions transverse to the distribution.

Confirmed the conjecture of Eliashberg and Mishachev in the analytic setting.

Abstract

Let $\xi$ 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 $\xi$. We formalise this as an asymptotic statement about finite jets of tangent curves. This solves, in the analytic setting, a conjecture of Y. Eliashberg and N.M. Mishachev regarding an earlier claim by M. Gromov about the microflexibility of the tangency condition. From these statements it follows, by an argument due to M. Gromov, that the $h$-principle holds for maps and immersions transverse to $\xi$.