An intrinsic and exterior form of the Bianchi identities

TL;DR

This paper presents a basis-independent, exterior calculus formulation of the Cartan structure equations and Bianchi identities, enhancing their flexibility and applicability across various geometric contexts.

Contribution

It introduces an elegant, basis-free approach to structure equations and Bianchi identities using exterior calculus, applicable to forms of arbitrary degree.

Findings

Unified formulation of structure equations and Bianchi identities

Demonstrated applicability to contact manifolds and foliations

Connected with classical mechanics via Cartan forms

Abstract

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms of arbitrary degree. We demonstrate the relationship with both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds, codimension one foliations and the Cartan form of classical mechanics are studied as examples of its flexibility and utility.