When we calculate Riemann Tensor for different curvature we have lots of components. However, there are many components that are zero. How can we argue, based on the symmetry of connection , that those elements are zero?
For example if I am calculating the Riemann Tensor of $S^2$ sphere, I get only one non zero component i.e. $R_{\phi,r,\phi}^{\theta}$ = $sin^2 {\theta}$ and other components are zero. So, How can I argue, without calculating that all other components are zero.
Edit:
(Dimension, No. of independent Riemann Components) = (2,1; 3,6 ; 4,20)
Best Answer
The number of independent components for the Riemann curvature tensor $R_{ijk\ell}$ for the Levi-Civita connection is greatly reduced because of symmetries. The last two indices $k\neq \ell$ have to be different, because of antisymmetry $$R_{ijk\ell}~=~-R_{ij\ell k}.$$ Interchange symmetry $$R_{ijk\ell}~=~R_{k\ell ij}$$ then fixes the first two indices $i\neq j$ to be different as well. In two dimensions, if the metric $g_{ij}$ is diagonal, then there is essentially only one non-zero possibility for $R_{ijk\ell}$ and $R^i{}_{jk\ell}$ up to symmetries.