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ck99
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Homework Statement
Show that Gij = Gji using the Riemann tensor identity (below)
Homework Equations
Gij = Rij - 1/2(gijR)
Rabcd + Rbcad + Rcabd = 0
R = gmrRmr
Rmr = Rmnrn
The Attempt at a Solution
I have tried to put the Ricci tensor and Ricci scalar (from the Gij equation) into full Riemann tensor form using the metric. For Gij I get
Rij = Rinjn = gnaRinja
gijR = gia gjb gab R = gia gjb Rab = gia gjb Ranbn = gia gjb gnd Ranbd
So Gij = ( gnaRinja - 1/2 gia gjb gnd Ranbd )
I have done the same for Gji but can't see how to link the two equations using the required Riemann identity. Is this the right approach?