smooth immersion

浸入

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

Translation evidence: Literature usage

Meaning in context

A smooth map between smooth manifolds whose differential is injective at every point of its domain.

光滑流形之间的光滑映射，在定义域的每一点，其切映射均为单射。

Example

The map $\mathbb R\to S^1$, $t\mapsto(\cos t,\sin t)$ is an immersion because $(-\sin t,\cos t)$ never vanishes, but it is not injective.

Usage and distinctions

Immersion is a differential-rank condition. An embedding also requires a homeomorphism onto the image with its subspace topology; immersion alone imposes neither global injectivity nor that topology.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.1.3, Definition 2.1.20 (immersion and submersion) · Checked 2026-10-14

Translation sources

1. Wang Jiajun — Peking University, 微分拓扑, Spring 2018 course page教学计划，2018-03-01：反函数定理、浸入；2018-03-20 作业内容 · Checked 2026-10-14

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