smooth vector bundle

向量丛

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Translation evidence: Literature usage

Meaning in context

A smooth projection over a smooth manifold with finite-dimensional real vector spaces of fixed rank as fibers and local product trivializations that are linear on each fiber; the changes between trivializations are smooth invertible linear maps.

光滑流形上的光滑投影，其纤维是固定有限维数的实向量空间，且有在每根纤维上线性的局部乘积平凡化；平凡化之间的变换为光滑的可逆线性映射。

Example

The quotient of [0,1]×R[0,1]\times\mathbb R by (0,t)∼(1,−t)(0,t)\sim(1,-t) is a real line bundle over S1S^1; its sign reversal makes it nontrivial.

Usage and distinctions

A fiber bundle may have nonlinear fibers and transition functions. In a vector bundle each fiber has linear operations, but vectors in different fibers cannot be added without extra choices.

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Definition sources

  1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.1.4, vector-bundle defining data and Definition 2.1.23 · Checked 2026-10-14

Translation sources

  1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要四、向量丛和主丛的联络：向量丛；PDF p. 77, printed p. 74 · Checked 2026-10-14
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