smooth submersion

淹没

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 surjective at every point of the domain; surjectivity of the map itself is not required.

光滑流形之间的光滑映射，在定义域每一点的切映射均为满射；不要求原映射整体为满射。

Example

The projection $(x,y)\mapsto x$ is a submersion $\mathbb R^2\to\mathbb R$. The map $x\mapsto x^2$ fails at zero because its differential there is zero.

Usage and distinctions

A submersion is locally a coordinate projection. A surjective smooth map can have critical points, so surjectivity of the map is a different condition.

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，2018-03-06：淹没、正则值原像定理；2018-03-20 作业内容 · Checked 2026-10-14

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