Selecting Indices for the Riemann Tensor

How do I know when computing the Riemann Tensor (in two dimensional) which indices to select? Consider the Riemann Tensor $$R^a_{bcd}$$ how do I know what values to take for $$a$$?

As an example, consider the 2 sphere with $$g_{ij}= r^2 , r^2 \sin^2\theta$$ in the coordinate frame $$(\theta , \phi )$$.

In the solution to the problem, I have seen that $$R^\phi_{\theta\theta\phi}$$ is selected but could I also choose for example to take $$R^\theta_{\theta\phi\theta}$$ or $$R^\theta_{\phi\phi\theta}$$ or $$R^\phi_{\theta\phi\theta}$$. Is there any way of deciding how do decide the upper index of the Riemann Tensor and does it matter which one I select or can I just pick any indices at random?

In two dimensions specifically, the symmetries of the Riemann tensor force it to have only one independent component. So, if you compute any non-vanishing component you will automatically know all the $$2^4 = 16$$ components of the tensor.
To be more specific, in two dimensions we have (if I'm not mistaken) $$R_{abcd} = \frac{R}{2}(g_{ac}g_{bd} - g_{a d}g_{b c}),$$ where $$R$$ is the Ricci scalar, and then you can just lift one index to get to the tensor you want.
• @missyclarke1998 Due to skew-symmetry in first two indices, we only need to know $R_{12cd}$. Due to skew-symmetry in the last two indices, we only need to know $R_{1212}=R_{2121}$, i.e $R_{\theta\phi\theta\phi}=R_{\phi\theta\phi\theta}$. If you then want the upper index version, just carry out the appropriate sum (but really I’d rather stick to the lowered version). Commented Apr 27 at 1:47