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  The Einstein equations for exterior forms

+ 1 like - 1 dislike
873 views

Let be given a manifold $M$, then we can consider the derivations over the exterior forms:

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

If $w$ is of even degree, $\Lambda_+ (M)$, we can define $w \wedge X$. The derivations are in isomorphism with $\Lambda_+ (M) \bigotimes \nabla (TM)$. The Koszul connection is defined by the two axioms ($w$ is even):

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; we suppose that we have an isomorphism with the dual. 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 (not necessarily even)? Can we construct a, so to say, tensor calculus for exterior forms?

asked Aug 31, 2019 in Mathematics by Antoine Balan (-80 points) [ revision history ]
edited Oct 19, 2019 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 - 1 dislike

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

answered Sep 6, 2019 by Antoine Balan (-80 points) [ revision history ]
edited Sep 17, 2019 by Antoine Balan

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