In studying Ricci flow I found that the Riemann Curvature tensor in a orthonormal frame can be written as $$Rm(\phi,\theta)=R_{abcd}\phi_{ab}\theta_{cd} , for \phi,\theta \in \Lambda^2(V).$$How this can be written? Where $\phi$ and $\theta$ are two forms

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    Mar 15, 2015 at 5:51

1 Answer 1


This is a standard result, directly implied by the symmetries of the Riemann tensor under exchange of the first and second (or third and fourth) indices. One interpretation is that the Riemann tensor then describes how derivatives in a plane (described by one of the two-forms) fail to commute and introduce terms from other planes (components of which are extracted by the other two-form).

Standard expressions for the Riemann tensor depend on decomposing one of the two-forms into one-forms. If the two-form is not simple, one can still write it as a linear combination of simple pieces.


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