Can we define flat connection on any given smooth manifold?

Question

For example, a sphere $S^2$ in $\mathbf{R}^3$ is apparently not flat with respect to the Euclidean connection, but can we define a flat connection and thus with affine charts on $S^2$?

Answer 1

There are topological obstructions to a vector bundle admitting a flat connection: most simply, by Chern-Weil theory the real Pontryagin classes of such a bundle must all vanish. So, for example, any closed $4$-manifold with nonzero signature, such as $\mathbb{CP}^2$, does not admit a flat connection.

Also by Chern-Weil theory, or by the Chern-Gauss-Bonnet theorem (which is stated on Wikipedia for the Levi-Civita connection but in fact holds for any connection), if an oriented vector bundle admits a flat connection then the real Euler class must also vanish, meaning that the Euler characteristic must be zero. So it follows that $S^2$ also does not admit a flat connection.