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A question that arises when dealing with vector bundles is that of the 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 ($\nabla_X\sigma=\nabla(X,\sigma)$)

  1. $\nabla_{X+Y}\sigma=\nabla_X\sigma+\nabla_Y\sigma$
  2. $\nabla_{fX}\sigma=f\nabla_X\sigma$
  3. $\nabla_X(\sigma+\tau)=\nabla_X\sigma+\nabla_X\sigma$
  4. $\nabla_X(f\sigma)=X(f)\sigma+f\nabla_X\sigma$

$\forall X$, $Y\in\chi(M)$, $\sigma$, $\tau\in \Gamma(E)$, $f\in C^\infty(M)$.