# Bivectors, Cartan Geometry and Curvature

1. Nov 9, 2009

### Orbb

I allow myself to repost some questions from the General Relativity section, as they may fit in better here:

I have 3 questions related to bivector space, the curvature tensor and Cartan geometry.

1) Because of its antisymmetric properties

$$R_{\mu\nu\alpha\beta}=-R_{\nu\mu\alpha\beta}$$ , $$R_{\mu\nu\alpha\beta}=-R_{\mu\nu\beta\alpha}$$ ,

the Riemann curvature tensor can be regarded as a second-rank bivector $$R_{AB}$$ in six-dimensional space (in case of spacetime dimension four). Due to the symmetry

$$R_{\mu\nu\alpha\beta}=R_{\alpha\beta\mu\nu}$$ ,

one can also conclude that $$R_{AB}=R_{BA}$$ . My question now is, which of the symmetry properties remain when extending Riemannian geometry to Cartan geometry with a non-symmetric Ricci-Tensor? Is it correct that one can still obtain a bitensor $$R_{AB}$$ , which then however is non-symmetric?

2) The six-dimensional space is of signature (+++---). Is there any analogue to Lorentz transformations in this space?

3) The metric $$g_{AB}$$ in bivector space can be constructed by

$$g_{\mu\nu\rho\sigma} = g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}$$ .

I guess from that one can derive a curvature tensor $$R_{ABCD}$$ for the six-dimensional space. Is that correct? And is there any interpretation for the bitensor representation $$R_{AB}$$ of $$R_{\mu\nu\alpha\beta}$$ ?