---
tag: KPQ96640ZJ
kind: concept
name: Morse function
msc: [57R70, "58E05", 58K05]
specializes: [0V33BNKZ9S]
sources:
  - url: https://en.wikipedia.org/wiki/Morse_theory
---

## Summary

A Morse function is a smooth real-valued function on a [[smooth manifold|F45VQ47GPV]] all of whose [[critical points|0YTCMFB4HC]] are nondegenerate.
This means that the [[Hessian|0693Y2METT]] at each critical point is invertible.
Near each critical point it has a simple quadratic normal form, given by the Morse lemma.
Morse functions exist on every smooth manifold and are generic among smooth functions.
They are the basic tool of Morse theory, which reads the topology of a manifold off the critical points of such a function.

## Explanations

### L1 human

Think of the height above the ground on a hilly landscape, or on a doughnut standing upright on a table.
Its critical points are the peaks, the pits and the passes, where the ground is flat.
A Morse function is one where each such flat spot is a clean peak, pit or saddle, not a flat plateau or a degenerate shape like a monkey saddle.
For the upright doughnut, the height has four critical points: a bottom, two saddles, and a top.