Connection form

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and specifically <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a connection form is a manner of organizing the data of a <a href="/facts/Connection_(mathematics)/KURsCmIu">connection</a> using the language of <a href="/facts/Moving_frame/3jNPLngL">moving frames</a> and <a href="/facts/Differential_form/T9gyHtMv">differential forms</a>.
Historically, connection forms were introduced by <a href="/facts/%C3%89lie_Cartan/Uvhkp66S">Élie Cartan</a> in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a <a href="/facts/Frame_bundle/Gugt3qyL">coordinate frame</a>, and so is not a <a href="/facts/Tensor/pNjFo5tV">tensorial</a> object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a>, a <a href="/facts/Connection_(principal_bundle)/UchSqJ0C">principal connection</a> is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a>, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In <a href="/facts/Physics/a7aH2y08">physics</a>, connection forms are also used broadly in the context of <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>, through the <a href="/facts/Gauge_covariant_derivative/sdAG6yRF">gauge covariant derivative</a>.
A connection form associates to each <a href="/facts/Basis_of_a_vector_space/89IPoN6c">basis</a> of a <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> of differential forms. The connection form is not tensorial because under a <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a>, the connection form transforms in a manner that involves the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> of the <a href="/facts/Atlas_(topology)/pjoya29t">transition functions</a>, in much the same way as the <a href="/facts/Christoffel_symbols/uCyd7RWX">Christoffel symbols</a> for the <a href="/facts/Levi-Civita_connection/16KGSQUY">Levi-Civita connection</a>. The main tensorial invariant of a connection form is its <a href="/facts/Curvature_form/LWp7DMDM">curvature form</a>. In the presence of a <a href="/facts/Solder_form/1zNxoKvT">solder form</a> identifying the vector bundle with the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a>, there is an additional invariant: the <a href="/facts/Torsion_(differential_geometry)/LVfDhPnW">torsion form</a>. In many cases, connection forms are considered on vector bundles with additional structure: that of a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> with a <a href="/facts/Lie_group/a9jCQyjF">structure group</a>.

Connection form open-in-new

Connection form