---
tag: 93MFWKENQ7
kind: definition
name: Global analysis
msc: [58C99, 58D15, 58J99]
defines: MTEW1M61QK
uses: [F45VQ47GPV, P4DZM975BV, GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Global_analysis
---

## Definition

The setting is the category of [[smooth manifolds|F45VQ47GPV]], smooth maps between them and smooth [[vector bundles|P4DZM975BV]] over them.
*Global analysis* is the branch of mathematics that studies, in this setting, the following objects.

1. [[Differential equations|GC5BDZ6QBC]] and differential operators whose unknowns are maps between manifolds or sections of bundles.
2. Variational problems on such maps and sections, and their critical points.
3. Spaces of maps between manifolds, seen as infinite-dimensional manifolds.

Its results concern the whole manifold, as opposed to results that hold in one coordinate chart.

## Remarks

The Hodge theorem is a result of global analysis.
It says that on a compact oriented Riemannian manifold each de Rham cohomology class has exactly one harmonic representative.
This is a statement about a differential operator, the Hodge Laplacian, and about the topology of the whole manifold.

A result that is not global is the local existence of solutions of an ordinary differential equation near a point.
It holds in one chart and says nothing about the whole manifold.

The name is also used for the analysis on infinite-dimensional manifolds of maps, which grew out of the same questions.