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In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950.〔Kobayashi (1957).〕 ==Soldering of a fibre bundle== Let ''M'' be a smooth manifold, and ''G'' a Lie group, and let ''E'' be a smooth fibre bundle over ''M'' with structure group ''G''. Suppose that ''G'' acts transitively on the typical fibre ''F'' of ''E'', and that dim ''F'' = dim ''M''. A soldering of ''E'' to ''M'' consists of the following data: # A distinguished section ''o'' : ''M'' → ''E''. # A linear isomorphism of vector bundles θ : T''M'' → ''o''−1V''E'' from the tangent bundle of ''M'' to the pullback of the vertical bundle of ''E'' along the distinguished section. In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism : from the tangent space of ''M'' at ''x'' to the (vertical) tangent space of the fibre at the point determined by the distinguished section. The form θ is called the solder form for the soldering. 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「solder form」の詳細全文を読む スポンサード リンク
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