# Riemann curvature tensor on a unit radius 2D sphere

• I
Summary:
How is it possible that non-zero components of the curvature tensor depend on theta
I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped.
I was surprised at this, because it implies that the curvature tensor tends to zero as we approach either pole. Being a sphere I thought that the curvature tensor would have the same value everywhere. How can this be?

## Answers and Replies

Ibix
2020 Award
According to my calculations, ##R^\phi{}_{\theta\phi\theta}=-1##. I'd check your algebra.

Dale
Mentor
2020 Award
Summary:: How is it possible that non-zero components of the curvature tensor depend on theta

Being a sphere I thought that the curvature tensor would have the same value everywhere. How can this be?
Because of the symmetry all of the invariants of the tensor must have the same value everywhere. But the coordinates are not symmetrical, so the components of the tensor in those coordinates may be non-constant.

epovo and Ibix
Ibix