So im trying to calculate the riemann tensor for $S^2$ for the metric $ds^2= d\theta ^2 +sin\theta ^2 d\phi$ and i have already calculated the Chrystoffel Symbols using the Euler-Lagrange Equations, my question is if there is any easier way or less tidious way to compute the Riemann Tensor than do the the sixteen possible combinations, since i am working with the Levi-Civita connection there might be some symmetries that aid to my calculations, i understand that i can use the explicit formula to calculate the Tensor but its just seems that is going to take a long time. I guess this question can be generalized for when working with the levi-civita connection when we want to calculate the riemann tensor is there something we can do to aid us? Thanks in advance.
Calculating the Riemann Tensor
riemannian-geometry
Best Answer
Well after this time i have figured it out , so we know the Chrystoffel symbols and we have the formula for the riemann tensor so we just to substitute them in the formula and there are alot of symmetries can be noticed so we really just need to calculate 2 values of the riemann tensor in certain elements of the basis of the tangent space and we have the problem resolved.