Metric connection

Riemannian connectioncompatibility with the metriccompatibleconnection one-formmetric connectionsmetric-compatibilityYang-Mills connectionYang-Mills connections
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve.wikipedia
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Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

Connection (vector bundle)

connectionKoszul connectionlinear connection
Then, a connection D on E is a metric connection if:
The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).

Parallel transport

parallelparallel-transporttransported
Equivalently, \nabla is Riemannian if the parallel transport it defines preserves the metric g.
In (pseudo) Riemannian geometry, a metric connection is any connection whose parallel transport mappings preserve the metric tensor.

Connection form

connectionconnection one-formvertical subspace
Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.
One may then define a connection that is compatible with this bundle metric, this is the metric connection.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

Gluon field strength tensor

gluon field tensor
The notation presented here is that which is commonly used in physics; for example, it can be immediately recognizable as the gluon field strength tensor.
A more mathematically formal derivation of these same ideas (but a slightly altered setting) can be found in the article on metric connections.

Gauge theory

gauge groupgauge theoriesgauge symmetry
The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.
For example, it is sufficient to ask that a vector bundle have a metric connection; when one does so, one finds that the metric connection satisfies the Yang-Mills equations of motion.

Bundle metric

Riemannian bundle metricfibre metricmetric
Let define a bundle metric, that is, a metric on the vector fibers of E.

Torsion tensor

torsiontorsion-freetorsion form
A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.

Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
In this case, the bundle E is the tangent bundle TM of a manifold, and the metric on E is induced by a Riemannian metric on M.

Hodge star operator

Hodge dualHodge starcodifferential
However, once one does require compatibility, this metric connection defines an inner product, Hodge star, Hodge dual, and Laplacian, which are required to formulate the Yang-Mills equations.

Laplace operator

LaplacianLaplacian operatorscalar Laplacian
However, once one does require compatibility, this metric connection defines an inner product, Hodge star, Hodge dual, and Laplacian, which are required to formulate the Yang-Mills equations.

Section (fiber bundle)

sectionsectionsLocal section
Let \sigma,\tau be any local sections of the vector bundle E, and let X be a vector field on the base space M of the bundle.

Differential form

differential forms2-formtwo-form
Here d is the ordinary differential of a scalar function.

Endomorphism

endomorphismsEndofunctionendomorphism monoid
The latter is a function on the bundle of endomorphisms so that

Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold
Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.

Curvature form

Bianchi identityBianchi identitiescurvature
Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.

Atlas (topology)

atlascharttransition map
The above definitions also apply to local smooth frames as well as local sections.

Exterior algebra

exterior productexterior powerwedge product
Let \Gamma(E) denote the space of differentiable sections on E, let \Omega^p(M) denote the space of p-forms on M, and let be the endomorphisms on E.

Physics

physicistphysicalphysicists
The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.