If you like this content, you can help maintaining this website with a small tip on my tipeee page

Our aim is to get more familiar with the Riemann curvature tensor and to calculate its components for a two-dimensional surface of a sphere of radius r.

First let's remark that for a two-dimensional space such as the surface of a sphere, the Riemann curvature tensor has **only one not null independent component**.

Actually as we know from our previous article The Riemann curvature tensor part III: Symmetries and independant components, the first pair and last pair of indices must both consist of different values in order for the component to be (possibly) non-zero. Therefore, in two dimensional space where each indice could only take the values 0 and 1, the only possibility for each pair is to contain these distinct indices 0 and 1, which represent the coordinates θ and φ in polar coordinates.

So, over 2^4 = 16 components of the Riemann tensor in two-dimensional space, only **one component is independent and not null**.

If we choose this component to be R_{θφθφ} for example, then we can easily deduce the three other not null components by using the Riemann tensor symmetries:

Therefore, we only have to calculate the first term R_{θφθφ}

_{}

Indeed, we recall from our article The Riemann curvature tensor for the surface of a sphere that the spacetime interval on the surface of a sphere of radius r in polar coordinates is:

**ds ^{2} = r^{2}dθ^{2} + r^{2}sin^{2}θdΦ^{2}**

So that we get as the corresponding metric g_{ij}:

which means that g_{θφ}=0 and that g_{θθ}=r^{2}

As the expression of the Riemann tensor as deduced in The Riemann curvature tensor part II: derivation from the geodesic deviation is given by

If we substitute the indices for R^{θ}_{φθφ },the above equation becomes

We now sum over dummy indice *m* to give

Now it is just a matter of calculating all these connection coefficients. We are lucky as we have already calculated them in our previous article Christoffel symbol exercise: calculation in polar coordinates part II; we have found that the eight Christoffel symbols at a given point on the surface of the sphere were:

We can therefore simplify our Riemann tensor expression to

So that finally

##### Ricci tensor

We can now find the Ricci tensor

So we get by summing over indices a and b

And finally the last two components of the Ricci tensor:

##### Ricci scalar

The last quantity to calculate is the Ricci scalar R = g^{ab}R_{ab}

We sum over the a and b indices to give

meaning the Ricci scalar decreases as the radius increase and tends to zero for large radii.