---
tag: 47096P24SB
kind: concept
name: Morse–Bott function
msc: [57R70, "58E05"]
specializes: [0V33BNKZ9S]
sources:
  - url: https://en.wikipedia.org/wiki/Morse_theory
---

## Summary

A Morse–Bott function is a smooth function whose critical points form submanifolds, and which is nondegenerate in the directions normal to these submanifolds.
It generalizes a [[Morse function|KPQ96640ZJ]], whose critical submanifolds are points.
Raoul Bott introduced the notion for functions with symmetry, whose critical points come in families.
An example is the energy on spaces of paths in a symmetric space.

## Explanations

### L1 human

Think of a ring-shaped doughnut lying flat on a table, and take the height as the function.
The lowest points form a whole circle, and so do the highest points, so the critical points are not isolated.
Still, in the direction across each circle the height curves up or down in a clean way.
Such a function is Morse–Bott: degenerate only along its critical circles and nondegenerate across them.
A simpler example is $f(x, y) = x^2$ on the plane, whose critical points form the line $x = 0$.