Graduate Anti-self Dual Part (2,2) Riemann Curvature Tensor

[abhinavabhatt]

TL;DR

Riemann curvature tensor has two pairs of anti-symmetric index. Is the double of Riemann tensor is Dual ? That how to define the anti self dual part of Riemann tensor.?

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.

 

[martinbn]

Not sure what the question is! You say you have a problem with a definition, but you haven't given a definition! Are you trying to figure out what the definition should be?! Where does all this come from and what is it for?

The Riemann tensor has two pairs of indices and you can use either of them to define a dual. Usually they are called the left and the right dual. If the metric is Einstein then they are equal.

 

[PeterDonis]

the problem is written in attached the file

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