Proving the symmetry property of Riemann curvature tensor

[Wan]

Homework Statement

Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and cyclic property.

Thanks!

2. Homework Equations
None

The Attempt at a Solution

I tried subbing in the definition of the curvature tensor but it didn't work. I've already done this homework question but am just wondering if there are other methods to do it.

 

[jedishrfu]

I think most people would do it via the way you described and move on.

However, you might some ideas here on how to do it differently:

http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/RiemannCurvature.pdf

And here:

https://physics.stackexchange.com/questions/135617/riemann-curvature-tensor-symmetries-proof

and this old discussion on PF:

https://www.physicsforums.com/threads/riemann-curvature-tensor-symmetries-proof.770969/

and lastly:

http://www.physics.ucc.ie/apeer/PY4112/Curvature.pdf