Bivectors, Cartan Geometry and Curvature

[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 $$R_{\mu\nu\alpha\beta}$$ ?

Any answers highly appreciated!

Cheers

 

[pat_connell]

1) Yes, the Ricci tensor in Cartan geometry is non-symmetric, so when expressed as a bitensor it will be non-symmetric. The symmetry properties of the Riemann curvature tensor still apply, however they may not be immediately apparent in the bitensor form. 2) The six-dimensional space does have its own Lorentz transformations, known as O(3,3) transformations. These are different from the usual Lorentz transformations in four dimensions, and can be used to transform between different bivector spaces.3) Yes, the metric in bivector space can be constructed by using the metric in four-dimensional spacetime. The resulting curvature tensor in six-dimensional space is known as the Weyl tensor. The bitensor representation of the Riemann curvature tensor can be interpreted as the components of the Weyl tensor in the same way that the components of the Riemann curvature tensor can be interpreted as components of the metric in four-dimensional spacetime.