cotangent space

余切空间

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

Translation evidence: Literature usage

Meaning in context

For a point p of a smooth real manifold M, the real dual vector space of its tangent space, consisting of real linear functions on tangent vectors at p.

对光滑实流形 $M$ 上一点 $p$，切空间 $T_pM$ 的实对偶空间，即该点切向量上的实线性函数所成的空间。

Example

On $\mathbb R^2$, $(dx)_p(a\partial_x+b\partial_y)=a$ and $(dy)_p(a\partial_x+b\partial_y)=b$; these covectors form a basis of $T_p^*\mathbb R^2$.

Usage and distinctions

Covectors evaluate vectors; they are not canonically tangent vectors. A chosen Riemannian metric supplies such an identification, which changes if the metric changes.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.3.1, dual vector bundles; §2.3.2, cotangent bundle and Example 2.3.5 · Checked 2026-10-14

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要二、向量场：余切空间；PDF p. 77, printed p. 74 · Checked 2026-10-14

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