High-Jet Relations of the Heat Kernel: Embedding Map and Applications

For any compact Riemannian manifold (M, g) and its heat kernel embedding map \(\psi_t :M \to l^2\) constructed in [BBG], we study the higher derivatives of \(\psi_t\) with respect to an orthonormal basis at x on M. As the heat flow time \(t \to 0_+\) , it turns out the limiting angles between these derivative vectors are universal constants independent on g, x and the choice of orthonormal basis. Geometric applications to the mean curvature and the Riemannian curvature are given. Some algebraic structures of the \(\infty\) -jet space of \(\psi_t\) are explored.