# Connections of degenerate metrics

1. Aug 24, 2010

### haushofer

I have a question about connections derived from degenerate metrics, like in Newton-Cartan.

The question is simply: can the transformation properties of the connection change if one considers connections derived from metrics which are degenerate?

One one hand, I would say that one can follow the usual GR-analysis, check how covariant derivatives of general vectors/covectors must transform and conclude how the connection must transform (with a inhomogeneous term).

On the other hand, in Newton-Cartan one has the metric conditions

$$\nabla_{\mu}\tau_{\nu} = 0, \ \ \ \ \nabla_{\rho}h^{\mu\nu}=0$$

where $h^{\mu\nu}\tau_{\nu}=0$; h plays the role of spatial metric, and tau the role of temporal metric.

So, do the transformation properties of the connection depend on the metrical structure?

2. Aug 26, 2010