In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.
A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of a vector bundle in a covariant manner. It also allows one to formulate a notion of a parallel section of a bundle in the direction of a vector: a section s is parallel along a vector X if . So a covariant derivative provides at least two things: a differential operator, and a notion of what it means to be parallel in each direction. An Ehresmann connection drops the differential operator completely and defines a connection axiomatically in terms of the sections parallel in each direction . 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. A section s is then horizontal (i.e., parallel) in the direction X if lies in a horizontal space. Here we are regarding s as a function from the base M to the fiber bundle E, so that is then the pushforward of tangent vectors. The horizontal spaces together form a vector subbundle of .
This has the immediate benefit of being definable on a much broader class of structures than mere vector bundles. In particular, it is well-defined on a general fiber bundle. Furthermore, many of the features of the covariant derivative still remain: parallel transport, curvature, and holonomy.
The missing ingredient of the connection, apart from linearity, is covariance.
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The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) 11-dimensional supergravity.
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. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G. A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection.
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.
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