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Orbb
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I have some 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
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