  The classical Bott-Samelson theorem states that if on a Riemannian manifold
all geodesics issuing from a certain point return to this point, then the
universal cover of the manifold has the cohomology ring of a compact rank one
symmetric space. This result on geodesic flows has been generalized to Reeb
flows and partially to positive Legendrian isotopies by
Frauenfelder-Labrousse-Schlenk. We prove the full theorem for positive
Legendrian isotopies.
