I mean, is it possible to have a connection on the 2-sphere with vanishing curvature but not vanishing torsion? In a more general sense, it is know that every Riemannian manifold has a Levi-Civita connection,is this true for the Weitzenböck connection?
It seems to me that a vector bundle on a simply connected manifold that has a flat connection must be trivial. (Parallel transport a frame at a base point to all other points.) But $S^2$ is far from parallelizable.