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?