In this Lecture 4, we use the Weitzenbock formula proved in the previous lecture to prove a generalized curvature dimension inequality, from which we will deduce Li-Yau type estimates for the horizontal heat kernel.

Part 1 covers sections 4.3, 4.4 and 5.1 in the Lecture Notes and Part 2 covers sections 5.2, 5.3 and 5.4.