Yes, I had worked some of that out for myself, starting with the fact that the sphere itself does not have any preferred directions. My goal is to arrive at the Einstein field equations, and I can see a much bigger headache coming when I tackle the stress energy tensor.
Good- we are getting somewhere at last. I didn't realize that the evaluation is done at a particular point and I have never seen that made clear in the textbooks. It still seems strange that it should be different at different places on the sphere. So - question answered, and my thanks to all!
Yes I know all that - you have just repeated what it says in my textbook, but it still doesn't answer my question. Since the radius of the sphere is 1, that makes this a numerical example, so theta should have a value, but what value is it, and why?
I mean the Riemann curvature tensor which as you say has 4 indices. The calculation for the unit sphere is standard and is given in many textbooks, presumably because it is the simplest possible case.
The Riemann curvature of a unit sphere is sine-squared theta, where theta is the usual azimuthal angle in spherical co-ordinates, and this is shown in many textbooks. But since a sphere is completely specified by its radius, then as far as I can see its curvature should be a function of its...