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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. Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory.
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his ...
The Riemannian connection or Levi-Civita connection [9] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in ...
In mathematics, and especially differential geometry and gauge theory, a connection 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.
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.
In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence of the curvature of the connection.
An Ehresmann connection drops the differential operator completely and defines a connection axiomatically in terms of the sections parallel in each direction (Ehresmann 1950). Specifically, an Ehresmann connection singles out a vector subspace of each tangent space to the total space of the fiber bundle, called the horizontal space .
In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.