Non-subanalyticity of sub-Riemannian Martinet spheres (Q2721622)

A sub-Riemannian structure \((M,D,g)\) is a triple, where \(M\) is a finite dimensional manifold, \(D\) is a distribution on \(M\), and \(g\) is a metric on \(D\). A curve \(C\) on \(M\) is called horizontal if it is tangent to \(D\). The author studies the problem of minimizing the length of \(C\) joining two points \(x_0\) and \(x_1\) a metric of the form \(g = a dx^2 + b dy^2\) where \(a\) is a function of \(y\) and \(b\) is a function of both \(x\) and \(y\). He proves the following main theorem: If \(a\) is a non-constant, then the sub-Riemannian sphere \(S(0,r)\) with small radii are not subanalytic.