The Cartan structure equations

Abstract

Abstract This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the torsion as a function of the connection forms and establishing the torsion differential 2-forms. It then turns to the curvature forms drawn from Chapter 23 and Cartan’s second structure equation, along with the curvature 2-forms. It also studies the Levi-Civita connection. The components of the Riemann tensor are then studied, with a Riemannian manifold, or a metric manifold with a torsion-less connection. The Riemann tensor of the Schwarzschild metric are finally discussed.