Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 19; Problem P19.6.
Here’s another example of the Riemann tensor in a 2-d coordinate system. The tensor is
As usual, we need the Christoffel symbols, which we can get by comparing the two forms of the geodesic equation. These equations are
The metric is
so and . For the two coordinates, 2 gives us
Dividing through by the coefficient of the second derivative in the second equation case gives:
Comparing with 3 we get
with all other Christoffel symbols equal to zero.
The only independent Riemann tensor component in 2-d is :
Any non-zero component indicates that the space is curved, so this metric represents a curved space.