# The Einstein equations for exterior forms

+ 1 like - 0 dislike
290 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
edited Oct 19, 2019

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"?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.