Given a manifold
M
M
and a proper sub-bundle
Δ
⊂
T
M
\Delta \subset TM
, we investigate homotopy properties of the horizontal free loop space
Λ
\Lambda
, i.e., the space of absolutely continuous maps
γ
:
S
1
→
M
\gamma :S^1\to M
whose velocities are constrained to
Δ
\Delta
(for example: legendrian knots in a contact manifold).
In the first part of the paper we prove that the base-point map
F
:
Λ
→
M
F:\Lambda \to M
(the map associating to every loop its base-point) is a Hurewicz fibration for the
W
1
,
2
W^{1,2}
topology on
Λ
\Lambda
. Using this result we show that, even if the space
Λ
\Lambda
might have deep singularities (for example: constant loops form a singular manifold homeomorphic to
M
M
), its homotopy can be controlled nicely. In particular we prove that
Λ
\Lambda
(with the
W
1
,
2
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
π
k
(
Λ
)
≃
π
k
(
M
)
⋉
π
k
+
1
(
M
)
\pi _k(\Lambda )\simeq \pi _k(M) \ltimes \pi _{k+1}(M)
for all
k
≥
0.
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
,
Δ
)
(M, \Delta )
is a compact sub-Riemannian manifold, each non-trivial homotopy class in
π
1
(
M
)
\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
,
Δ
)
(M, \Delta )
is a compact, contact manifold, then every sub-Riemannian metric on
Δ
\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
Λ
\Lambda
).