Cut locus and heat kernel at Grushin points of 2 dimensional almost Riemannian metrics

Abstract

This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y). After a short introduction, I first give a description of the local cut and conjugate loci at a Grushin point q (where Dq has dimension 1 and Dq = TqM) that makes appear that the cut locus may have an angle at q. In a second time I describe the local cut and conjugate loci at a Riemannian point x in a neighborhood of a Grushin point q. Finally, applying results of [6], I give the asymptotics in small time of the heat kernel p_t(x,y) for y in the same neighborhood of q.