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  The Einstein equations for exterior forms

+ 1 like - 0 dislike
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Let be given a manifold $M$, then we can consider the derivations over the exterior forms of degree $2k$ $\Lambda_+ (M)$:

$X(a \wedge b)=X(a) \wedge b+ a \wedge X(b)$

If $w$ is in $\Lambda_+ (M)$, we can define $w \wedge X$. The derivations are in isomorphism with $\Lambda_+ (M) \bigotimes TM$. The Koszul connection is:

i) $\nabla_X (w \wedge s)= X(w) \wedge s + w \wedge \nabla_X (s)$

ii) $ \nabla_{w \wedge X} (s)= w \wedge \nabla_X (s)$

We can define a riemannian metric: $g(X,Y)=g(Y,X)$ and $g(w \wedge X,Y)=w \wedge g(X,Y)$ with $X,Y$ two derivations. So we can define the Levi-Civita connection and the Riemann and Ricci curvature which are tensors over $M$. In this formalism, the Einstein equations for exterior forms seem to exist. Are they interesting in physics? Can we extend in supersymmetry to any exterior forms?

asked Aug 31 in Mathematics by Antoine Balan (85 points) [ revision history ]
edited 3 days ago by Antoine Balan

I do not understand your question. The Riemann curvature exists, so there exist Einstein equations for it. But what do you mean by "Einstein equations for exterior forms"?

1 Answer

+ 0 like - 0 dislike

I replace the smooth functions by the even exterior forms, is it coherent and interesting ?

answered Sep 6 by Antoine Balan (85 points) [ revision history ]
edited 3 days ago by Antoine Balan

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