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水平丛.
在数学微分几何领域,给定
- π:E→M,
是光滑流形 M 上一个光滑纤维丛,则 E 的铅直丛 VE 是切丛 TE 的一个子丛,由与 E 在 M 上的纤维相切的切向量组成。一个水平丛(horizontal bundle)则是特别地选取 TE 的一个子丛使其为 VE 的补丛,换句话说,在每个纤维给出一个补空间。
完全一般地,水平丛概念是表述纤维丛上埃雷斯曼联络的一种途径。但这个概念经常用于更确定的情形。
更具体的,如果 e ∈ E 满足
- π(e)=x ∈ M,
则在 e 处铅直空间(vertical space) VeE 是纤维 Ex 穿过 e 的切空间 Te(Ex)。一个水平丛则确定了一个水平空间(horizontal space)HeE 使得 TeE 是 VeE 与 HeE 的直和。
如果 E 是一个主 G-丛则水平丛通常要求为 G-等变;更多细节参见联络。特别地,当 E 是标架丛便是这种情形,标架丛是流形的切空间的所有标架,而 G = GLn。
- Kobayashi, Shoshichi and Nomizu, Katsumi. Foundations of Differential Geometry, Vol. 1. Wiley-Interscience. 1996 (New edition). ISBN 0471157333.
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