Cartan's Structural Equations for Singular Manifolds

TL;DR

This paper extends Cartan's structural equations to singular manifolds with degenerate metrics, enabling the study of geometric properties near singularities by constructing analogous connection and curvature forms.

Contribution

It introduces a canonical method to define geometric objects for radical stationary fundamental tensors, generalizing Cartan's equations to singular manifolds.

Findings

Constructs geometric objects for degenerate metrics

Derives analogs of Cartan's structural equations

Provides a compact Koszul formula version

Abstract

The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate. But if the fundamental tensor is allowed to be degenerate, there are some obstructions in constructing the geometric objects normally associated to the fundamental tensor. Also, local orthonormal frames and coframes no longer exist, as well as the metric connection and its curvature operator. This article shows that, if the fundamental tensor is radical stationary, we can construct in a canonical way geometric objects, determined only by the fundamental form, similar to the connection and curvature forms of Cartan. In particular, if the fundamental tensor is non-degenerate, we obtain the usual connection and curvature forms of Cartan. We write analogs of Cartan's first and second structural equations. As a byproduct we will find a compact version of the Koszul formula.