This article is a research into relations between analytical properties and differential invariants for the geometry of a contact sub-Riemannian manifold of dimension \(3\) using the asymptotic expansion of the heat kernel associated to the sub-Riemannian heat equation on such manifolds.
The author considers a complete sub-Riemannian manifold \(M\) of dimension \(3\) with a contact structure \(\omega\), i.e., \(\omega\) is a one form on \(M\) where \(\omega\wedge d\omega\) is a non-vanishing \(3\)-form on \(M.\) Denoting \(\mathcal{D}:=\mathrm{ker}(\omega)\) the smooth distribution on \(M\) with constant rank \(2\), it is assumed that \(g_q\) is a Riemannian metric on \(\mathcal{D}_q\) depending smoothly on \(q\in M.\) This gives rise to a metric on \(M\) determined by the Carnot-Caratheodory distance \[ d(q_0, q_1)=\mathrm{inf}\{\int_0^T|\dot{\gamma}(t)|\mathrm{d}t: \gamma: [0, T]\rightarrow M \text{ Lip.,} \gamma(0)=q_0, \gamma(T)=q_1, \dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)}\}. \] Locally, the sub-Riemannian structure is defined by the orthonormal frame \(\{f_1, f_2\}\) satisfying the conditions: \[ \mathcal{D}=\mathrm{span}\{f_1, f_2\}, \quad g(f_i, f_j)=\delta_{ij}, \quad d\omega(f_1, f_2)=1. \] In addition, the Reed vector field associated to the contact structure is the unique vector field \(f_0\) where \(\omega(f_0)=1\) and \(d\omega(f_0, \cdot)=0.\) Denote the functions \(c_{ij}^k\) on \(M\) the structure constants for the Lie algebra generated by \(f_1, f_2, f_3\): \[ [f_1, f_0]=c_{01}^1f_1+c_{01}^2f_2, \quad [f_2, f_0]=c_{02}^1f_1+c_{02}^2f_2, \quad [f_2, f_1]=c_{12}^1f_1+c_{12}^2f_2+f_0. \] There are two differential invariants \(\chi, \kappa\) in contact geometry of dimension \(3\), locally giving by the structure constants. In particular, \[ \kappa=f_2(c_{12}^1)-f_1(c_{12}^2)-(c_{12}^1)^2-(c_{12}^2)^2+\frac{c_{01}^2+c_{02}^1}{2}. \] The spectral geometry of the contact manifold is captured by the sub-Laplacian \(\Delta_f\), which in dimension \(3\) has the form \[ \Delta_f=f_1^2+f_2^2+c_{12}^2f_1-c_{12}^1f_2. \] The heat kernel \(p\) is the Schwarz kernel of the heat operator \(e^{t\Delta_f}\) associated to the sub-Laplacian \[ e^{t\Delta_f}\varphi(x)=\int_M p(t, x, y)\varphi(y)\mathrm{d}y, \qquad \varphi\in C_0^{\infty}(M). \] The main result of this paper relates the first two terms of the small time asymptotic expansion of the heat kernel \(p\) to the geometric invariants of the manifold \(M\): \[ p(t, x, y)\sim\frac{1}{16t^2}[1+\kappa(x)t+o(t)]\quad\text{as}\quad t\to0. \] This result provides a concrete example of contact geometry in dimension \(3\) in understanding relations of geometric invariants to the analysis on manifolds.