# Linear connection

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In the mathematical field of [differential geometry](/source/differential_geometry), the term '''linear connection''' can refer to either of the following overlapping concepts:
* a [connection on a vector bundle](/source/connection_(vector_bundle)), often viewed as a differential operator (a ''Koszul connection'' or ''[covariant derivative](/source/covariant_derivative)'');
* a [principal connection](/source/connection_(principal_bundle)) on the [frame bundle](/source/frame_bundle) of a manifold or the induced connection on any [associated bundle](/source/associated_bundle) — such a connection is equivalently given by a [Cartan connection](/source/Cartan_connection) for the [affine group](/source/affine_group) of [affine space](/source/affine_space), and is often called an [affine connection](/source/affine_connection).

The two meanings overlap, for example, in the notion of a linear connection on the [tangent bundle](/source/tangent_bundle) of a manifold.

In older literature, the term ''linear connection'' is occasionally used for an [Ehresmann connection](/source/Ehresmann_connection) or [Cartan connection](/source/Cartan_connection) on an arbitrary fiber bundle,<ref>{{springerEOM|id=c/c025140|title=Connection (on a fibre bundle)|author=[Ülo Lumiste](/source/%C3%9Clo_Lumiste)}}</ref> to emphasise that these connections are "linear in the horizontal direction" (i.e., the [horizontal bundle](/source/horizontal_bundle) is a ''vector'' subbundle of the tangent bundle of the fiber bundle), even if they are not "linear in the vertical (fiber) direction". However, connections which are not linear in this sense have received little attention outside the study of [spray structures](/source/Spray_(mathematics)) and [Finsler geometry](/source/Finsler_geometry).

==References==
{{reflist}}

Category:Connection (mathematics)

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