Distortion coefficients and exponential map in sub-Riemannian geometry

The aim of this thesis is to explore the fields of sub-Riemannian and metric geometry. We compute the distortion coefficients of the α-Grushin plane. These distortion coefficients are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a synthetic curvature bound for the $\alpha$-Grushin plane is proposed. We then prove a version of Warner's properties for the sub-Riemannian exponential map. The regularity property is established by considering sub-Riemannian Jacobi fields while the continuity property follows from studying the Maslov index of Jacobi curves. We show how this implies that the exponential map of the Heisenberg group is not injective in any neighbourhood of a conjugate vector. In the appendix, we prove that the curvature-dimension for negative effective dimension fails to hold in any strict and complete sub-Riemannian manifold