Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I always find a connection that has both Riemann curvature and torsion tensors identically zero?


I've attempted to simply for the Christoffel symbols, but couldn't make headway in the equations.



Does every manifold have a flat connection?
Milnor proved in [On the existence of a connection with curvature zero, Comm. Math. Helv. v 32] that bundles over a surface of genus g has flat connections iff its Euler class is less than g by an absolute value (see also Wood, Bundles with totally disconnected structure group). Sullivan in "A generalization of Milnor's inequality ...Comm. Math. Helv. v. 51" find a finite upper bound for the Euler class of a R^n-bundle with the affine connection over manifold M^n (the number of n-simplices in the triangulation of M^n). Hope, this might help. 


