- #1
Oxymoron
- 870
- 0
I have
[tex]R^{ij} = c(e^i \wedge e^j)[/tex]
as my curvature 2-form, (and c represents "constant" curvature). I would like to contract to form the Ricci 1-forms, [itex]P_a[/itex], which in turn, would allow me to write out the Ricci tensor as
[tex]\mbox{Ric} = P_a \otimes e^a[/tex]
and eventually, the Einstein tensor.
My question is, how would I go about contracting my 2-form? I think it is fairly easy and I am just missing something. I mean there is a big difference between saying that [itex]R^i{}_i[/itex] is the Ricci 1-form because I contracted the 2-form, [itex]R_{ij}[/itex] and writing down exactly what just happened.
[tex]R^{ij} = c(e^i \wedge e^j)[/tex]
as my curvature 2-form, (and c represents "constant" curvature). I would like to contract to form the Ricci 1-forms, [itex]P_a[/itex], which in turn, would allow me to write out the Ricci tensor as
[tex]\mbox{Ric} = P_a \otimes e^a[/tex]
and eventually, the Einstein tensor.
My question is, how would I go about contracting my 2-form? I think it is fairly easy and I am just missing something. I mean there is a big difference between saying that [itex]R^i{}_i[/itex] is the Ricci 1-form because I contracted the 2-form, [itex]R_{ij}[/itex] and writing down exactly what just happened.