Calculate the elements of the Riemann tensor

[zardiac]

Homework Statement

Compute 21 elements of the Riemann curvature tensor in for dimensions. (All other elements should be able to produce through symmetries)

Homework Equations

$R_{abcd}=R_{cdab}$
$R_{abcd}=-R_{abdc}$
$R_{abcd}=-R_{bacd}$

The Attempt at a Solution

I don't see how 21 elements can be enough,
for example
abcd
0000
1111
2222
3333
1110
1100
1000
2220
2200
2000
3330
3300
3000
0012
0120
0023
0230
1112
1122
1212
2223
2233
2323

I am already up to 24 elements and I am not even sure if this is enough to derive all the other ones. Am I missunderstanding something?
According to http://mathworld.wolfram.com/RiemannTensor.html it should be possible, but I really don't see it right now.
Any hints would be appreciated.

 

[clamtrox]

There are 20 components of the tensor which can be non-zero. For example R₀₀₀₀ is certainly 0 (why?)

 

[andrien]

see 8th heading in the following,it will guide you how there are only 20 independent components and not 256 as it might be think(4×4×4×4)
http://www.mathpages.com/rr/appendix/appendix.htm

 

[zardiac]

Ah of course, there are some elements that are zero. I don't know why I didn't checked that first. Thanks for quick replies, I think I got it right now :)

 

[paranormal]

I would suggest using the symmetries and properties of the Riemann curvature tensor to reduce the number of calculations needed. For example, the symmetry R_{abcd} = -R_{bacd} can reduce the number of elements to be calculated by half. Similarly, the symmetries R_{abcd} = R_{cdab} and R_{abcd} = -R_{abdc} can further reduce the number of elements to be calculated. Additionally, the properties R_{abcd} = -R_{abdc} and R_{abcd} = -R_{bacd} can also be used to reduce the number of elements. By using these symmetries and properties, it may be possible to calculate the remaining elements needed to get a total of 21 elements. However, it may also be necessary to use other techniques or methods to fully calculate all the elements of the Riemann tensor. Further research and exploration may be needed in order to fully understand and calculate the Riemann tensor.