Universal curvature identities

TL;DR

This paper investigates universal curvature identities, applying heat equation methods to reprove classical theorems and identities related to curvature, Euler-Lagrange equations, and the Gauss-Bonnet theorem.

Contribution

It introduces new proofs and derivations of key curvature identities and theorems using universal curvature identities and heat equation techniques.

Findings

Established the Gauss-Bonnet theorem via heat equation methods.

Provided a new proof of Kuz'mina and Labbi's Euler-Lagrange equations.

Derived the Euh-Park-Sekigawa identity anew.

Abstract

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss-Bonnet theorem using heat equation methods, to give a new proof of a result of Kuz'mina and Labbi concerning the Euler-Lagrange equations of the Gauss-Bonnet integral, and to give a new derivation of the Euh-Park-Sekigawa identity.