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once again I'm stuck on something I am quite certain is silly, but here it goes. My confusion pertains to the equation

[itex]Ric=R^{a}\otimes e_{a}[/itex]

where [itex]Ric[/itex] is the Ricci tensor, [itex]R^{a}[/itex] is the Ricci 1-form and [itex]e_{a}[/itex] are the elements of an orthonormal basis.

Now, lets say for arguments sake that [itex]a=0,1,2[/itex] and I have a Ricci 1-form that looks something like this (What I'm actually trying to work out is a lot larger but follows a similar pattern)

[itex]R^{a}=\left[ \begin{array}{c} Ae_{0} + Be_{1} \\ Be_{0} - Ae_{1} \\ e_{2} \end{array} \right][/itex]

where [itex]A[/itex] and [itex]B[/itex] are constants. The next step would be to take the tensor product of [itex]R^{a}[/itex] and [itex]e_{a}[/itex] and this is where the problem lies. My instinct would be to treat this as an outer product so you end up with something like

[itex]R^{a}\otimes e_{a}=\left[ \begin{array}{ccc} (Ae_{0} + Be_{1})e_{0} & (Ae_{0} + Be_{1})e_{1} & (Ae_{0} + Be_{1})e_{2} \\ (Ae_{0} - Be_{1})e_{0} & (Ae_{0} - Be_{1})e_{1} & (Ae_{0} - Be_{1})e_{2} \\ e_{2}e_{0} & e_{2}e_{1} & e_{2}e_{2} \end{array} \right][/itex]

But that seems to be ignoring the sum over [itex]a[/itex] (or is this the operation it implies?) and more importantly, I really doubt there should be multiplication between the elements, i.e does

[itex](Ae_{0} + Be_{1})e_{0}[/itex]

imply

[itex](Ae_{0} + Be_{1})\otimes e_{0}[/itex]

or

[itex](Ae_{0} + Be_{1})\wedge e_{0}[/itex]

As said, this is a really silly thing to be stuck with and probably means that I've missed(read not paid attention to) something really basic so any help would be very much appreciated.

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# Ricci tensor from Ricci 1-form

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