Differential Forms on Riemannian (Lorentzian) and Riemann-Cartan Structures and Some Applications to Physics

TL;DR

This paper reviews differential forms on various geometric structures, corrects a recent formula related to torsion, and discusses applications in physics, providing detailed exercises for students and experts.

Contribution

It offers a detailed correction of a recent formula involving the exterior covariant derivative of torsion forms and applies differential geometry tools to physical contexts.

Findings

Corrects a recent formula for the exterior covariant derivative of torsion forms

Provides detailed exercises on differential forms in different geometric settings

Discusses applications of differential forms in physics

Abstract

In this paper after recalling some essential tools concerning the theory of differential forms in the Cartan, Hodge and Clifford bundles over a Riemannian or Riemann-Cartan space or a Lorentzian or Riemann-Cartan spacetime we solve with details several exercises involving different grades of difficult. One of the problems is to show that a recent formula appearing in the literature for the exterior covariant derivative of the Hodge dual of the torsion 2-forms is simply wrong. We believe that the paper will be useful for students (and eventually for some experts) on applications of differential geometry on physical problems. A detailed account of the issues discussed in the paper appears in the table of contents.