You live at some spot in an n dimensional Riemannian manifold and have a convenient coordinate system. Each of the indexes i, j, k and l take on one of n values; there are n^{4} such combinations and as many values for R^{i}_{jkl}. For each such set of values R^{i}_{jkl} can be evaluated by the following simple procedure. Travel in direction k until coordinate x^{k} is increased by dx^{k}. (You choose dx^{k}, but see guidance at end.) The other coordinates will not have changed. Next travel indirection l for distance dx^{l}. Now travel backwards in the k direction but a distance −dx^{k}. Then once more in the direction l for distance −dx^{l}. You will arrive back near to where you left. That was just practice.
Now travel the same route again but carry along a vector V that initially points in the direction j with magnitude dx^{j}.
(You choose dx^{j} too.)
At each step in the journey do not turn V that you carry, even though the coordinates may themselves turn which will cause the coordinate description of V to change.
This would be a big effect if you were walking near the North pole.
There are several ways that the meaning of this can be defined which we do not explore here.
When you get back compare the V you carried with a copy that you left behind.
The difference between the vector you started with and the one you brought back will be a new vector with components D^{1}, ...D^{n}.
R^{i}_{jkl} = D^{i}/(dx^{j}dx^{k}dx^{l}).
You must choose the size of dx^{j} etc. and this is a practical tradeoff—too small and D will be too small to measure, too large and your curvature result won’t really be local. If your manifold is mathematically defined then you should let the dx’s tend to zero and take the limit.
If R_{ijkl} = g_{iβ}R^{β}_{jkl} then
R_{ijkl} = −R_{jikl} = −R_{ijlk} = R_{klij}
R_{ijkl} + R_{ikjl} + R_{iljk} = 0.
These constraints allow only n^{2}(n^{2} − 1)/12 degrees of freedom in an n dimensional space.
Here is a table:
n | n^{4} | n^{2}(n^{2} − 1)/12 |
2 | 16 | 1 |
3 | 81 | 6 |
4 | 256 | 20 |
5 | 625 | 50 |
Γ^{i}_{jk} tells how fast the coordinates turn. It is thus about the coördinate system, and only indirectly about the space.