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)