How to know which the independent components of the Riemann tensor are before calculating them? I want to know how to figure out the non-vanishing and independent components of the Riemann Tensor before computation so I don't need to calculate all of them.
For example, a $3x3$ metric would give 27 Riemann tensor components but only 6 are independent, how do I go straight for them?
 A: You can exploit symmetries. The number 6 you quote comes from taking them into account.
The way I'd approach it is to try and start "populating" the indices of the tensor, in order, while keeping the symmetries into account.
Say we are calling the coordinates 1, 2 and 3.
The first index can be anything, let's set it to 1.
So, we are looking at $R_{1***}$. The second index cannot be 1 (that would vanish), so we can set it to either 2 or 3. Let's start with 2: we have $R_{12**}$.
What can the third index be? 1 works, no symmetry makes it vanish, so let's go with it. The last one cannot be 1, again by the antisymmetry of the last two indices, so we set it to 2.
Our first nonvanishing component is therefore $R_{1212}$.
You may notice that the two index pairs 12 and 12 are necessarily two distinct indices; so our options for exchanging them are 12, 13, and 23 (while to get the components with 21, 31, and 32 one can simply invert the sign).
This seems to lead to eight more independent components: $R_{1213}$, $R_{1223}$, $R_{1312}$, $R_{1313}$, $R_{1323}$, $R_{2312}$, $R_{2313}$, $R_{2323}$.
However, three of the nine components we have written out are not independent: specifically, there is symmetry in switching the first and last index pairs, so $R_{1213} = R_{1312}$, $R_{1223} = R_{2312}$, $R_{1323} = R_{2313}$. This leaves us with six independent components, which we know to be the correct amount, so we are done.
This line of reasoning also works with more dimensions, although it quickly gets quite lengthy to do by hand.
