This dissertation contains two research directions. In the first direction, we deduce explicit expressions of the subelliptic heat kernels on three sub-Riemannian model spaces: the Cauchy-Riemann sphere, the anti-de Sitter space and the Quaternionic sphere. From these explicit subelliptic heat kernels we then derive several by products: the Green function of the conformal sub-Laplacian, the small-time estimates of the subel- liptic heat kernels, and the sub-Riemannian distance. The key point is to work in cylindrical coordinates that reflect the symmetries coming from the Hopf fibration of these model spaces.

In the second direction we study the extension of the Baudoin-Garofalo type curvature dimension inequality from the sub-Riemannian transversal symmetric setting to any contact manifold. In particular, the Sasakian condition is no longer assumed which leads to the appearance of new strongly non-linear term in the curvature dimension inequality. This new curvature dimension condition is then used to study several interesting aspects in geometry and analysis: The stochastic completeness of the heat semigroup, geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results), gradient bounds for the heat semigroup, and spectral gap estimates for the sub-Laplacian.