Many interesting hypoelliptic diffusion operators may be studied by introducing a wellchosen Riemannian foliation. In particular many sub-Laplacians on sub-Riemannians manifolds often appear as horizontal Laplacians of a foliation and many kinetic type hypoelliptic operators, like the Kolmogorov operators appear as the sum of the vertical Laplacian of a foliation and of a first order term. In a joint program with Nicola Garofalo it has been proved, under very general conditions, that if a

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... iptic operator satisfies a generalized curvature dimension inequality then many results are available. The applications of the generalized curvature dimension inequality have been covered in the first part of the course. In this second part we focus on the geometric framework in which this curvature-dimension estimate is available. In particular, we will prove that if a Riemannian foliation is totally geodesic, then under natural geometric conditions, the horizontal Laplacian satisfies the generalized curvature dimension inequality. In the last lecture, we will study some problems related to Kolmogorov type operators. Lecture 1: We will introduce the concept of Riemannian foliation and define the horizontal and vertical Laplacians. Basic theorems like the Bérard-Bergery-Bourguignon commutation theorem will be proved. Lecture 2: We will study explicit examples of Riemannian foliation with totally geodesic leaves that can be seen as model spaces. These examples are all associated to the Hopf fibration and natural generalizations of it. We will give explicit expressions for the horizontal heat kernels of these model sapces. Lecture 3: We will prove a transverse Weitzenböck formula for the horizontal Laplacian of a Riemannian foliation with totally geodesic leaves. As a first consequence of this Weitzenböck formula, we prove that if natural assumptions are satisfied, then the horizontal Laplacian satisfies the generalized curvature dimension inequality. As a second consequence, we will prove sharp lower bounds for the first eigenvalue of the horizontal Laplacian.