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Concept

Manifold

Geometry
Curved Surface with a Flat Coordinate Patch

A manifold is a topological space that locally resembles Euclidean space near each point, so that a manifold's global structure can be complicated even though every small neighbourhood around any point looks like ordinary flat space. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Origin Year
1854 1
Connections

Associated With

Bernhard Riemann, Mathematicians
Source Manifold (Wikipedia)
De Rham's Theorem, Theorems
Tensor, Concepts

Tensor fields defined at each point of a manifold, varying smoothly across it, are the standard way tensors extend beyond a single vector space into differential geometry.

Additional Source Tensor (Wikipedia)Lead section
Whitney Immersion Theorem, Theorems

Attributed To

Max Dehn, Mathematicians
Source Manifold (Wikipedia)
Poul Heegaard, Mathematicians
Source Manifold (Wikipedia)

In Branch

Algebraic Topology, Branches of Mathematics
Differential Geometry, Branches of Mathematics
Source Manifold (Wikipedia)
Topology, Branches of Mathematics
Source Manifold (Wikipedia)
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsRiemann biography, 1854 habilitation lecture
Quote, Riemann biography, 1854 habilitation lecture
Riemann's lecture Ueber die Hypothesen welche der Geometrie zu Grunde liegen, delivered on 10 June 1854, became a classic of mathematics.
View the Source
Manifold (Wikipedia)
Wikimedia Foundation
  • lead paragraph
    a manifold is a topological space that locally resembles Euclidean space near each point
  • Associated With: Bernhard Riemann, history section
    Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions.
  • Attributed To: Max Dehn, Lead
    Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn
  • Attributed To: Poul Heegaard, Lead
    Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn
View the Source
Tensor (Wikipedia)
Wikimedia FoundationAssociated With: Tensor, Lead section
Quote, Associated With: Tensor, Lead section
a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space
View the Source

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