I'm currently in the process of deriving the components of the Riemann Curvature Tensor for a 3-sphere using the Cartan Equations. The line element I'm starting with is:
$$ ds^2 = \frac{dr^2}{1-r^2/R^2} + r^2 d\phi^2 + r^2\sin^2(\phi) d\theta^2 $$
During this procedure, differentiation of these terms is necessary, but the $R$ term is actually defined as another term involving $r$ to simplify the original line element to derive the one above. Now, the crux of this whole question is:
Do I simply differentiate $R$ with respect to $r$, or must its original definition be taken into account? In other words, do I treat $R$ just like $r$?
I've consulted several other sources on this, and they appear to do just this, but I'm not particularly comfortable with this as $R$ is a term involving $r$, NOT $r$ itself!!
Any pointers would be greatly appreciated!!!