---
tag: MTEW1M61QK
kind: concept
name: Global analysis
msc: [58C99, 58D15, 58J99]
sources:
  - url: https://en.wikipedia.org/wiki/Global_analysis
same_as:
  wikidata: Q17017346
  wikipedia: Global analysis
---

## Summary

Global analysis is the study of differential equations, variational problems and analysis on [[smooth manifolds|F45VQ47GPV]] as whole objects.
It looks at properties that depend on the entire manifold and not only on one chart.
Its tools include [[differential operators on vector bundles|CGBX1N9V8H]], [[spaces of smooth maps|BW12J14X0Z]], [[critical point theory|9ZQC7ARWTT]] and [[index theory|675MAXBQN6]].
It links analysis with the topology of manifolds, for instance through Morse theory and the Atiyah–Singer index theorem.

## Explanations

### L1 human

Classical calculus studies functions near one point or on a piece of flat space.
Global analysis asks what happens on a curved space, such as a sphere or a torus, when the whole space is taken into account.
A typical question is how many solutions an equation has on the whole space, or how many critical points a function must have.
For example, every smooth function on a sphere has at least two critical points, a maximum and a minimum, because the sphere is compact.
The shape of the space thus forces facts about the analysis on it.