Curvature Questions, Please Help! 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!