smooth manifold

微分流形

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

Translation evidence: Literature usage

Meaning in context

A Hausdorff, second countable space locally homeomorphic to open subsets of a fixed Euclidean space, together with a smooth structure represented by charts whose transition maps are infinitely differentiable; boundary is excluded here.

具有 Hausdorff 性和第二可数性、局部同胚于固定维数欧氏空间开集，并配有坐标变换无穷次可微的微分结构的空间；本条不含边界。

Example

The unit sphere S2S^2 is smooth: near a point with z≠0z\ne0, solve z=±1−x2−y2z=\pm\sqrt{1-x^2-y^2} to obtain local coordinates.

Usage and distinctions

Local Euclidean topology alone does not choose a smooth structure. The union of two crossing coordinate axes fails even the manifold condition at the crossing.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§1.2.1, Definition 1.2.1 and Remark 1.2.2 · Checked 2026-10-14

Translation sources

  1. Peking University — 2022 mathematics teaching plans课程号00130190，微分流形，内容提要一：微分流形的定义；PDF p. 77, printed p. 74 · Checked 2026-10-14
Edit source / propose a PRReport a correction

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