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CARTAN'S STRUCTURAL EQUATIONS FOR DEGENERATE METRIC

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. On sin- gular semi-Riemannian manifolds, because the metric is allowed to be degenerate, there are some obstructions in constructing the geometric objects normally associated to the metric. We can no longer construct local orthonormal frames and coframes, or define a metric connection and its curvature operator. But we will see that if the metric is radical stationary, we can construct objects similar to the connection and curvature forms of Cartan, to which they reduce if the metric is non-degenerate. We write analogs of Cartan's first and second structural equations. As a byproduct we will find a compact version of the Koszul formula.