About: Metric connection

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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. This is equivalent to: * A connection for which the covariant derivatives of the metric on E vanish. * A principal connection on the bundle of orthonormal frames of E.

Attributes	Values
rdf:type	road
rdfs:label	Metrischer Zusammenhang (de) Metric connection (en) Метрическая связность (ru) Метрична зв'язність (uk)
rdfs:comment	Ein metrischer Zusammenhang beziehungsweise ein mit der Metrik kompatibler Zusammenhang ist ein mathematisches Objekt aus der Differentialgeometrie. Es handelt sich um einen Spezialfall eines Zusammenhangs. (de) Метрическая связность ― линейная связность в векторном расслоении снабжённом билинейной метрикой в слоях, при которой параллельное перенесение вдоль произвольно кусочно гладкой кривой в сохраняет метрику, то есть скалярное произведение векторов остаётся постоянным при их параллельном перенесении. (ru) Метрична зв'язність ― лінійна зв'язність у векторному розшаруванні з білінійною метрикою в шарах, при якій паралельне перенесення вздовж довільно кусочно гладкої кривої в зберігає метрику, тобто скалярний добуток векторів залишається сталим при їх паралельному перенесенні. (uk) 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. This is equivalent to: * A connection for which the covariant derivatives of the metric on E vanish. * A principal connection on the bundle of orthonormal frames of E. (en)
dcterms:subject	Riemannian geometry Connection (mathematics)
Wikipage page ID	4794503 (xsd:integer)
Wikipage revision ID	1068777208 (xsd:integer)
Link from a Wikipage to another Wikipage	Electromagnetism Volume element Lie derivative Connection form Critical point (mathematics) Mathematics Gauge theory Tangent bundle Christoffel symbols Einstein notation Electromagnetic tensor Endomorphism Fundamental theorem of Riemannian geometry General relativity Gluon field strength tensor Connection (principal bundle) Connection (vector bundle) Contorsion tensor Hodge dual Plug and chug Levi-Civita connection Lie algebra Physics Alternating form Euler–Lagrange equation Parallel transport Gravitation (book) Kaluza–Klein theory Riemann curvature tensor Orthonormal frame Pseudo-Riemannian manifold Vector potential Cotangent bundle Covariant derivative Atlas (topology) Riemannian geometry Connection (mathematics) Laplacian Torsion tensor Differential form Bundle metric Inner product Metric tensor Linear form Orthogonal group Yang–Mills equations Section (fiber bundle) Skew-symmetric matrix Vertical and horizontal bundles Vector bundle Hodge star Yang–Mills action Riemann curvature Bianchi's second identity Casimir operator Field strength tensor Christoffel symbol Connection coefficient Connection one-form Smooth frame Local frame
Link from a Wikipage to an external page	http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103858479%7C
sameAs	Metric connection Metric connection Metric connection Metric connection Metric connection Metric connection
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_arXiv dbt:Cite_journal dbt:Main dbt:Redirect dbt:Reflist dbt:Short_description dbt:Sup dbt:Use_American_English dbt:Nowarp dbt:Riemannian_geometry dbt:Tensors dbt:Manifolds
has abstract	Ein metrischer Zusammenhang beziehungsweise ein mit der Metrik kompatibler Zusammenhang ist ein mathematisches Objekt aus der Differentialgeometrie. Es handelt sich um einen Spezialfall eines Zusammenhangs. (de) 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. This is equivalent to: * A connection for which the covariant derivatives of the metric on E vanish. * A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection is a ; there is a unique such which is torsion free, the Levi-Civita connection. 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. Another special case of a metric connection is a , which satisfies the Yang–Mills equations of motion. Most of the machinery of defining a connection and its curvature can go through without requiring any compatibility with the bundle metric. 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. (en) Метрическая связность ― линейная связность в векторном расслоении снабжённом билинейной метрикой в слоях, при которой параллельное перенесение вдоль произвольно кусочно гладкой кривой в сохраняет метрику, то есть скалярное произведение векторов остаётся постоянным при их параллельном перенесении. (ru) Метрична зв'язність ― лінійна зв'язність у векторному розшаруванні з білінійною метрикою в шарах, при якій паралельне перенесення вздовж довільно кусочно гладкої кривої в зберігає метрику, тобто скалярний добуток векторів залишається сталим при їх паралельному перенесенні. (uk)
gold:hypernym	Connection
prov:wasDerivedFrom	wikipedia-en:Metric_connection?oldid=1068777208&ns=0

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