# Curvature Questions,

1. Nov 16, 2008

### JasonJo

1. The problem statement, all variables and given/known data
1) Prove that if M is locally symmetric (i.e. the Riemann tensor is constant), connected and 2 dimensional, then M has constant sectional curvature.
2) Prove that if M has constant (sectional) curvature, then M is a locally
symmetric space.

2. Relevant equations
R(X, Y) Z = constant along any geodesic, i.e. it is a parallel vector field.

3. The attempt at a solution
For the first part:
Since M is two dim, the sectional curvature coincides with the actual curvature. Why do we need that M is connected? Am I suppose to use that the sectional curvature (hence the Riemann curvature) does not change along geodesics?

* What exactly does constant sectional curvature mean? Does it mean that the sectional curvature K does not depend on the 2-dimensional space and that it does not change along any curve?

For the second part:
Not quite sure about this part. I see the idea and the picture, but what is the first step?

Thanks guys!