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

TL;DR

This paper investigates the higher derivatives of the heat kernel embedding map on compact Riemannian manifolds, revealing universal limiting angles and exploring geometric and algebraic structures as the heat flow parameter approaches zero.

Contribution

It introduces the analysis of higher derivatives of the heat kernel embedding, uncovering universal geometric constants and algebraic structures in the jet space.

Findings

Universal constants for angles between derivatives as t approaches 0

Connections between derivatives and geometric quantities like curvature

Exploration of algebraic structures in the jet space

Abstract

For any compact Riemannian manifold $(M,g)$ and its heat kernel embedding map $psi_t$ from M into $l^2$ 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 $t$ 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 $psi_t$ are explored.