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 Rθφθφ 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 Rθφθφ

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:

ds2 = r22 + r2sin2θdΦ2

So that we get as the corresponding metric gij:

 which means that gθφ=0 and that gθθ=r2

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 Rθφθφ ,the above equation becomes

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