Georgia Institute of Technology College of Sciences

Assuming some "compatibility" conditions between a Riemannian metric and a contact structure on a 3-manifold, it is natural to ask whether we can use methods in global geometry to get results in contact topology. There is a notion of compatibility in this context which relates convexity concepts in those geometries and is well studied concerning geometry questions, but is not exploited for topological questions. I will talk about "contact sphere theorem" due to Etnyre-Massot-Komendarczyk, which might be the most interesting result for contact topologists.