High-jet relations of the heat kernel embedding map and applications

Abstract

For any compact Riemannian manifold $(M,g)$ and its heat kernel embedding map $ps{i}_t$ from M into $l^2$ constructed in [BBG], we study the higher derivatives of $ps{i}_t$ with respect to an orthonormal basis at $x$ on $M$. As the heat flow time $t$ goes 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 infinite jet space of $ps{i}_t$ are explored.

28 pages, related references on random functions are added

References in corpus (2)

• Random Morse functions and spectral geometry
• Isometric embeddings via heat kernel