Connection (mathematics)

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In geometry, the notion of a connection makes precise the idea of transporting data
along a curve or family of curves in a parallel and consistent manner.

There are various kinds of connections in modern geometry,
depending on what sort of data one wants to transport.

Connections are of central importance in modern geometry in large part
because they allow a comparison between the local geometry at one point and
the local geometry at another point.

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

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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)

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A Cartan connection is a way of formulating some aspects of connection theory using differential forms and Lie groups.

An Ehresmann connection is a connection in a fibre bundle or a principal bundle by
specifying the allowed directions of motion of the field.

A Koszul connection is a connection which defines directional derivative for sections of a vector bundle
more general than the tangent bundle.

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