orientation of a smooth manifold

光滑流形的定向

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

Translation evidence: Literature usage

Meaning in context

On a smooth real n-manifold, an equivalence class of nowhere-vanishing smooth n-forms, where two forms are equivalent when one is a positive smooth function times the other; such a class exists exactly when the manifold is orientable.

在光滑实 $n$ 维流形上，处处非零的光滑 $n$ 形式的等价类；两形式相差一个正的光滑函数因子时等价。存在这样的等价类当且仅当流形可定向。

Example

On $\mathbb R^2$, $dx\wedge dy$ and $e^x dx\wedge dy$ define the same orientation; $-dx\wedge dy$ defines the opposite orientation.

Usage and distinctions

Orientability is existence of an orientation, whereas orientation is a choice. An orientation fixes signs in integration without selecting the lengths or areas supplied by a metric.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§3.4.2, Definition 3.4.4 and the volume-form equivalence relation immediately following it · Checked 2026-10-14

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要二：光滑流形的定向；PDF p. 77, printed p. 74 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.