The Bott-Samelson theorem for positive Legendrian isotopies

TL;DR

This paper extends the Bott-Samelson theorem to positive Legendrian isotopies, establishing conditions under which the universal cover of a manifold has the cohomology of a rank one symmetric space, generalizing previous results on Reeb flows.

Contribution

The paper proves the full Bott-Samelson theorem for positive Legendrian isotopies, completing a partial generalization from Reeb flows.

Findings

Universal cover has the cohomology of a compact rank one symmetric space under specified conditions

Extension of Bott-Samelson theorem to positive Legendrian isotopies

Completes previous partial results in the field

Abstract

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.