---
tag: T9H6PT9FHE
kind: concept
name: Morse theory on Hilbert manifolds
msc: ["58E05", 58B20, 57R70]
sources:
  - url: https://en.wikipedia.org/wiki/Palais–Smale_compactness_condition
---

## Summary

Morse theory on Hilbert manifolds extends classical Morse theory from compact manifolds to functions on infinite-dimensional [[Hilbert manifolds|R8DMMW0E9M]].
Richard Palais and Stephen Smale replaced compactness of the manifold by a compactness condition on the function, the [[Palais–Smale condition|4724KRNHAT]].
With it, sublevel sets change only at critical values, and passing a nondegenerate critical point of finite index $\lambda$ attaches a $\lambda$-cell.
The main application is the study of geodesics as critical points of the energy on spaces of paths and loops.

## Explanations

### L1 human

Classical Morse theory reads the shape of a compact manifold off the critical points of a function on it.
Many natural problems live on infinite-dimensional spaces, such as the space of all paths between two points, where the function is the energy of a path.
Such spaces are not compact, so a sequence that looks more and more like a critical point might escape to infinity.
The Palais–Smale condition rules this out, and then the classical theory goes through.
For example, critical points of the energy on paths from $p$ to $q$ are the geodesics from $p$ to $q$, and their indices count conjugate points.