On heat kernels of a class of degenerate elliptic operators

In this article we study the geometry induced by a class of secondorder subelliptic operators. This class contains degenerate elliptic and hypoelliptic operators (such as the Grushin operator and the Baouendi-Goulaouic operator). Given any two points in the space, the number of geodesics and the lengths of those geodesics are calculated. We find modified complex action functions and show that the critical values of these functions will recover the lengths of the corresponding geodesics. We also find the volume elements by solving transport equations. Then heat kernels for these operators are obtained.