Morse function

Morse函数

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

Translation evidence: Literature usage

Meaning in context

A smooth real-valued function on a smooth manifold whose Hessian at each critical point is a nonsingular symmetric bilinear form; functions with no critical points are included.

光滑流形上的光滑实值函数，其每个临界点处的 Hessian 对称双线性型均非退化；无临界点的函数也满足此定义。

Example

On $T^2=(\mathbb R/2\pi\mathbb Z)^2$, $f(\theta,\phi)=\cos\theta+2\cos\phi$ has four nondegenerate critical points with indices $0,1,1,2$.

Usage and distinctions

An isolated critical point may still have singular Hessian, as for $x^4$ at zero. Morse functions do not require distinct critical values, and their definition does not require a Riemannian metric.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Ralph L. Cohen, Kevin Iga and Paul Norbury — Topics in Morse TheoryIntroduction, printed pp. v–vi, Hessian, nondegenerate critical points and Morse-function definition; Chapter 5, Theorem 5.3 · Checked 2026-10-15

Translation sources

1. Nankai University — 数学科学学院本科生论坛-83讲题目：Morse理论简介；摘要中“流形上一类特殊的函数（Morse函数）” · Checked 2026-10-15

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