Math 2.0

Global analysis, analysis on manifolds Definition

Global analysis

Builds on Smooth manifoldVector bundlePartial differential equation

Definition

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

  1. Differential equations 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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Partial differential equation, concept, no evidence levelPartial differential …conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptVector bundle, concept, no evidence levelVector bundleconceptGlobal analysis, definition, no evidence levelGlobal analysisdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Partial differential equation concept
  2. Smooth manifold concept
  3. Vector bundle concept
  4. Global analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Global_analysis
About this page
Tag93MFWKENQ7
URIm2://93MFWKENQ7
Kinddefinition
WritingW0 stub
MSC58C99, 58D15, 58J99
DefinesGlobal analysis
UsesSmooth manifold, Vector bundle, Partial differential equation

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