# The Einstein equations for exterior forms

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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?

edited Oct 19

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"?

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