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| In [[mathematics]], a '''metric connection''' is a [[connection (vector bundle)|connection]] in a [[vector bundle]] ''E'' equipped with a [[metric (vector bundle)|metric]]<!--Red link until someone wants to write an appropriate article. [[metric tensor]] isn't right.--> for which the [[inner product]] of any two vectors will remain the same when those vectors are [[parallel transport]]ed along any curve. Other common equivalent formulations of a metric connection include:
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| * A connection for which the [[connection (vector bundle)|covariant derivative]]s of the metric on ''E'' vanish.
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| * A [[connection (principal bundle)|principal connection]] on the bundle of [[orthonormal frame]]s of ''E''.
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| A special case of a metric connection is the [[Levi-Civita connection]]. Here the bundle ''E'' is the [[tangent bundle]] of a manifold. In addition to being a metric connection, the Levi-Civita connection is required to be [[torsion tensor|torsion free]].
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| ==Riemannian connections==
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| An important special case of a metric connection is a '''Riemannian connection'''. This is a connection <math>\nabla</math> on the [[tangent bundle]] of a [[pseudo-Riemannian manifold]] (''M'', ''g'') such that <math>\nabla_X g = 0</math> for all vector fields ''X'' on ''M''. Equivalently, <math>\nabla</math> is Riemannian if the [[parallel transport]] it defines preserves the metric ''g''.
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| A given connection <math>\nabla</math> is Riemannian if and only if
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| :<math>Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ) </math> | |
| for all vector fields ''X'', ''Y'' and ''Z'' on ''M'', where <math>Xg(Y,Z)</math> denotes the derivative of the function <math>g(Y,Z)</math> along this vector field <math>X</math>.
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| The [[Levi-Civita connection]] is the [[torsion tensor|torsion-free]] Riemannian connection on a manifold. It is unique by the [[fundamental theorem of Riemannian geometry]].
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| ==External links==
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| *[http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103858479 a pdf about this]
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| [[Category:Connection (mathematics)]]
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| [[Category:Riemannian geometry]]
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| {{differential-geometry-stub}}
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