Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics

Abstract:

Given a manifold $M$ and a proper sub-bundle $\Delta \subset TM$, we investigate homotopy properties of the horizontal free loop space $\Lambda$, i.e., the space of absolutely continuous maps $\gamma :S^1\to M$ whose velocities are constrained to $\Delta$ (for example: legendrian knots in a contact manifold).

In the first part of the paper we prove that the base-point map $F:\Lambda \to M$ (the map associating to every loop its base-point) is a Hurewicz fibration for the $W^{1,2}$ topology on $\Lambda$. Using this result we show that, even if the space $\Lambda$ might have deep singularities (for example: constant loops form a singular manifold homeomorphic to $M$), its homotopy can be controlled nicely. In particular we prove that $\Lambda$ (with the $W^{1,2}$ topology) has the homotopy type of a CW-complex, that its inclusion in the standard free loop space (i.e., the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently that its homotopy groups can be computed as $\pi _k(\Lambda )\simeq \pi _k(M) \ltimes \pi _{k+1}(M)$ for all $k\geq 0.$

In the second part of the paper we address the problem of the existence of closed sub-Riemannian geodesics. In the general case we prove that if $(M, \Delta )$ is a compact sub-Riemannian manifold, each non-trivial homotopy class in $\pi _1(M)$ can be represented by a closed sub-Riemannian geodesic.

In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if $(M, \Delta )$ is a compact, contact manifold, then every sub-Riemannian metric on $\Delta$ carries at least one closed sub-Riemannian geodesic. This result is based on a combination of the above topological results with the delicate study of an analogue of a Palais-Smale condition in the vicinity of abnormal loops (singular points of $\Lambda$).