We say a connection $\nabla$ on a vector bundle $\pi \colon E \to M$ is flat if the distribution on $E$ is integrable. The Frobenius Theorem tells us that a connection is flat if and only if it is locally trivial.

The curvature of a connection gives a quantitative measurement of far away the connection is from being flat. Later in the course we will see that for Riemannian manifolds, the curvature does indeed correspond to the physical meaning of the word. For instance, the sphere $S^m$ with its Euclidean metric is “positively” curved.