Why does 2 dimensional homogeneous riemannian manifold have constant sectional curvature? I have no idea. Any hints to get me thinking?
On a 2-dimensional manifold there is only one 2-dimensional direction therefore there is only one sectional curvature at a point, namely the Gaussian curvature. By Gauss's theorema egregium the Gaussian curvature is invariant under isometries. Therefore if the manifold is homogeneous, i.e., there is an isometry moving any point to any other, the conclusion follows.