The Standard Laplace Operator

TL;DR

This paper explores the properties of the standard Laplace operator, a generalization of classical operators on differential forms, emphasizing its functorial nature and commutation with various differential operators on geometric vector bundles.

Contribution

It demonstrates that the standard Laplace operator commutes with a broad class of natural differential operators, extending its functorial properties in geometric analysis.

Findings

Standard Laplace operator generalizes classical operators.

It commutes with many natural first order differential operators.

Includes examples illustrating these properties.

Abstract

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stressing the functorial aspects of the standard Laplace operator $\Delta$ with respect to the category of geometric vector bundles we show that the standard Laplace operator commutes not only with all homomorphisms, but also with a large class of natural first order differential operators between geometric vector bundles. Several examples are included to highlight the conclusions of this article.