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

TL;DR

This paper studies the geometric structure of 2D almost Riemannian manifolds, focusing on cut and conjugate loci at Grushin points, and analyzes the small-time asymptotics of the heat kernel in these regions.

Contribution

It provides a detailed description of the local cut and conjugate loci at Grushin points and their influence on heat kernel asymptotics in 2D almost Riemannian structures.

Findings

Cut locus may have an angle at Grushin points.

Local cut and conjugate loci are characterized near Grushin points.

Small-time heat kernel asymptotics are derived near Grushin points.

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.