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In mathematics, a connection on a fiber bundle 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. If the fiber bundle is a vector bundle, then the notion of parallel transport must be linear. Such a connection is equivalently specified by a ''covariant derivative'', which is an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Connections in this sense generalize, to arbitrary vector bundles, the concept of a linear connection on the tangent bundle of a smooth manifold, and are sometimes known as linear connections. Nonlinear connections are connections that are not necessarily linear in this sense. Connections on vector bundles are also sometimes called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them . ==Formal definition== Let ''E'' → ''M'' be a smooth vector bundle over a differentiable manifold ''M''. Denote the space of smooth sections of ''E'' by Γ(''E''). A connection on ''E'' is an ℝ-linear map : such that the Leibniz rule : holds for all smooth functions ''f'' on ''M'' and all smooth sections σ of ''E''. If ''X'' is a tangent vector field on ''M'' (i.e. a section of the tangent bundle ''TM'') one can define a covariant derivative along ''X'' : by contracting ''X'' with the resulting covariant index in the connection ∇ (i.e. ∇''X''σ = (∇σ)(''X'')). The covariant derivative satisfies the following properties: : Conversely, any operator satisfying the above properties defines a connection on ''E'' and a connection in this sense is also known as a covariant derivative on ''E''. 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「Connection (vector bundle)」の詳細全文を読む スポンサード リンク
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