A question that arises when dealing with vector bundles is that of the

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*parallel transport*. Just like happens in Riemannian geometry, there is no canonical way to move vectors along the manifold, but this requires an additional structure, a*connection*. But the more direct way to describe it is by a function that, given a direction and a field along this direction, points out the (infinitesimal) deviation with respect to the parallel transport.A

**connection $\nabla$ in a vector bundle $E$**is a map $\nabla:\chi(M)\times \Gamma(E)\longrightarrow \Gamma(E)$ satisfying ($\nabla_X\sigma=\nabla(X,\sigma)$)- $\nabla_{X+Y}\sigma=\nabla_X\sigma+\nabla_Y\sigma$
- $\nabla_{fX}\sigma=f\nabla_X\sigma$
- $\nabla_X(\sigma+\tau)=\nabla_X\sigma+\nabla_X\sigma$
- $\nabla_X(f\sigma)=X(f)\sigma+f\nabla_X\sigma$