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

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The discussion centers on the challenges of defining the anti-self dual part of the Riemann curvature tensor, particularly in relation to the Weyl curvature tensor, which has pairs of anti-symmetric indices. Participants express confusion over the lack of a clear definition provided for the duality concepts being discussed. The Riemann tensor's duals, referred to as left and right duals, can be defined using its two pairs of indices, with the equality of these duals noted in the context of an Einstein metric. There is a call for clearer communication and the use of LaTeX for mathematical expressions to enhance understanding. Overall, the thread highlights the need for precise definitions and clarity in discussing advanced curvature concepts.
abhinavabhatt
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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.
 

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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.
 
abhinavabhatt said:
the problem is written in attached the file
This is not acceptable. Please use the PF LaTeX feature to post equations directly in the thread. There is a "LaTeX Guide" button at the lower left of the post window.
 

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