- #1
Tangent87
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Homework Statement
I am trying to show that for a vector field Va which satisfies [tex]V_{a;b}+V_{b;a}[/tex] that [tex]V_{a;b;c}=V_eR^e_{cba}[/tex] using just the below identities.
Homework Equations
[tex]V_{a;b;c}-V_{a;c;b}=V_eR^e_{abc}[/tex](0)
[tex]R^e_{abc}+R^e_{bca}+R^e_{cab}=0[/tex] (*)
[tex]V_{a;b}+V_{b;a}=0[/tex] (**)
The Attempt at a Solution
So far I have:
[tex]V_eR^e_{cba}=-V_eR^e_{acb}-V_eR^e_{bac}=V_{a;b;c}-V_{a;c;b}+V_{b;c;a}-V_{b;a;c}=2V_{a;b;c}+V_{b;c;a}-V_{a;c;b}[/tex]
Which gives:
[tex]V_eR^e_{cba}-V_{a;b;c}=V_eR^e_{abc}-V_{c;b;a}[/tex]
I want to say that this implies [tex]V_{a;b;c}=V_eR^e_{cba}[/tex] but I can't justify why both sides of the above equation must be zero. Can anyone see what else I can do? Thanks