Quantitative Darboux theorems in contact geometry

This paper begins the study of relations between Riemannian geometry and contact topology on (2n+1)-manifolds and continues this study on 3-manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact (2n+1)-manifold (M,\xi ) that can be embedded in the standard contact structure on \mathbb{R}^{2n+1}, that is, on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form \alpha for \xi . In dimension 3, this further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curve techniques to provide a lower bound for the radius of a PS-tight ball.