Contraction between Levi-Civita symbol and Riemann tensor

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Discussion Overview

The discussion revolves around proving the contraction of the Levi-Civita symbol with the Riemann tensor, specifically the expression εμνρσ Rμνρσ = 0. The scope includes theoretical aspects of tensor symmetries and identities in differential geometry.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to prove the expression εμνρσ Rμνρσ = 0.
  • Another suggests using the symmetries of the Riemann tensor to approach the proof.
  • A different participant expresses uncertainty about the specific symmetry relation Rμνρσ = Rμ(νρ)σ, indicating that if true, it would simplify their understanding.
  • Another participant proposes starting with the first Bianchi identity and contracting indices with the Levi-Civita symbol, suggesting that rearranging indices could lead to the desired result.
  • A participant acknowledges the use of the Bianchi identity Rμ[νρσ]=0 as a helpful step in their reasoning.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to prove the expression. Multiple competing views and methods are presented, indicating that the discussion remains unresolved.

Contextual Notes

There are limitations regarding the assumptions about the symmetries of the Riemann tensor and the specific forms of the identities mentioned. The discussion does not clarify whether certain symmetries hold universally.

mhob
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How to proof that
εμνρσ Rμνρσ =0 ?

Thanks.
 
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Last edited by a moderator:
ShayanJ said:
Use the symmetries of the Riemann tensor!
I'm not sure Rμνρσ = Rμ(νρ)σ or not? If so, the problem solved for me.

Thanks.
 
I thought those symmetries would help, but it seems they don't. Start with the first Bianchi identity and contract all the indices with a Levi-Civita symbol. Rearranging the indices of the Levi-Civita symbol in two of the terms will give you what you want.
 
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Likes   Reactions: mhob
OK, I know use the Bianchi identity
Rμ[νρσ]=0
Thanks!
ShayanJ!
 

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