tangent bundle

切丛

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a smooth n-manifold, the disjoint union of all its tangent spaces, with projection sending a tangent vector to its point and smooth local trivializations obtained by differentiating coordinate charts; it is a real vector bundle of rank n.

对光滑 $n$ 维流形，将各点切空间作不交并，以切向量所在点为投影，并由坐标卡的微分给出光滑局部平凡化，所得秩为 $n$ 的实向量丛。

Example

The field $(-y,x)$ on $S^1$ never vanishes and gives $TS^1\cong S^1\times\mathbb R$ by $(p,a)\mapsto a(-p_2,p_1)$.

Usage and distinctions

The rank is n, but the total-space dimension is 2n. Local product descriptions do not imply a global product description; the tangent bundle of $S^2$ is nontrivial.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.1.2, The tangent bundle; §2.1.4, Vector bundles · Checked 2026-10-14

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00137914，第3–4周：切丛与余切丛的各种定义及其统一；PDF p. 78, printed p. 75 · Checked 2026-10-14

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