[share]

In mathematics, and especially differential geometry and gauge theory,
a connection on a fiber bundle is a device that defines a notion of parallel transport on
the bundle; that is, a way to "connect" or identify fibers over nearby points.

The most common case is that of a linear connection on a vector bundle,
for which the notion of parallel transport must be linear.

A linear connection is equivalently specified by a covariant derivative, an operator that
differentiates sections of the bundle along tangent directions in the base manifold,
in such a way that parallel sections have derivative zero.

Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on
the tangent bundle of a Riemannian manifold, which gives a standard way to
differentiate vector fields.
Nonlinear connections generalize this concept to bundles whose fibers are not
necessarily linear.

https://en.wikipedia.org/wiki/Connection_(vector_bundle)