Homotopy Properties of Horizontal Loop Spaces and Applications to Closed Sub-riemannian Geodesics

Given a manifold M and a proper sub-bundle ∆ ⊂ TM , we study homotopy properties of the horizontal base-point free loop space Λ, i.e. the space of absolutely continuous maps γ : S → M whose velocities are constrained to ∆ (for example: legendrian knots in a contact manifold). A key technical ingredient for our study is the proof that the base-point map F : Λ → M (the map associating to every loop its base-point) is a Hurewicz fibration for the W 1,2 topology on Λ. Using this result we show that, even if the space Λ 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 Λ (with the W 1,2 topology) has the homotopy type of a CW-complex, that its inclusion in the standard base-point free loop space (i.e. the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently its homotopy groups can be computed as πk(Λ) ≃ πk(M) ⋉ πk+1(M) for all k ≥ 0. These topological results are applied, in the second part of the paper, to the problem of the existence of closed sub-riemannian geodesics. In the general case we prove that if (M,∆) is a compact sub-riemannian manifold, each non trivial homotopy class in π1(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,∆) is a compact, contact manifold, then every sub-riemannian metric on ∆ carries at least one closed sub-riemannian geodesic. This result is based on a combination of the above topological results with a delicate study of the Palais-Smale condition in the vicinity of abnormal loops (singular points of Λ).