How Rⁱ_jkl Measures the Curvature of Space

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 n⁴ such combinations and as many real values for Rⁱ_jkl. For each such set of values Rⁱ_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 D¹, ...Dⁿ.
Rⁱ_jkl = Dⁱ/(dx^jdx^kdx^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_iβR^β_jkl then
R_ijkl = −R_jikl = −R_ijlk = R_klij
R_ijkl + R_iljk + R_iklj = 0. These constraints allow only n²(n² − 1)/12 degrees of freedom in an n dimensional space. Here is a table:

n	n4	n2(n2 − 1)/12
1	1	0
2	16	1
3	81	6
4	256	20
5	625	50

Why do you use such inefficient notation? Isn’t there a better way to describe 4D curvature than writing 256 numbers when you need only 20? Because the tensor notation simplifies all the useful equations. We find tricks to avoid writing 256 numbers, even in computers.

Γⁱ_jk tells how fast the coordinates turn. It is thus about the coördinate system, and only indirectly about the space.