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In mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. Other common equivalent formulations of a metric connection include: * 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 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 free. ==Riemannian connections== An important special case of a metric connection is a Riemannian connection. This is a connection on the tangent bundle of a pseudo-Riemannian manifold (''M'', ''g'') such that for all vector fields ''X'' on ''M''. Equivalently, is Riemannian if the parallel transport it defines preserves the metric ''g''. A given connection is Riemannian if and only if : for all vector fields ''X'', ''Y'' and ''Z'' on ''M'', where denotes the derivative of the function along this vector field . The Levi-Civita connection is the torsion-free Riemannian connection on a manifold. It is unique by the fundamental theorem of Riemannian geometry. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「metric connection」の詳細全文を読む スポンサード リンク
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