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Affine connection

In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of…

History, Motivation and history & Further properties

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Overview

Motivation and history

Formal definition as a differential operator

Parallel transport for affine connections

Formal definition on the frame bundle

Affine connections as Cartan connections

Further properties

Surface theory revisited

Advanced semantic analysis

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Map overview Semantic statistics

Affine connection

Nodes140
Edges139
Triples93
Avg. degree1.99
Density0.014286
Components1

How this topic connects Entity context

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Affine connection

Top relations

related to Elementary properties · 13
Affine connection → Any, Conversely, Gamma, If, In, It, More, Rn, The, TM, Xx, XY, Yx
related to Motivation from tensor calculus · 11
Affine connection → Before, Bernhard Riemann, Christoffel, Correction, Elwin Bruno Christoffel, Euclidean, Gregorio Ricci-Curbastro, The, These, This, Tullio Levi-Civita
related to Affine connections as Cartan connections · 8
Affine connection → Affine, Cartan, Cartan's, FM, From, However, In, Indeed
related to Development · 8
Affine connection → An, As, Ct, In, Intuitively, Then, Tx0M, TxtM
related to Parallel transport for affine connections · 8
Affine connection → An, By, Comparison, Intuitively, Let, Such, Then, YX
related to Surface theory revisited · 7
Affine connection → From, Furthermore, If, It, Levi-Civita, R3, Riemannian
related to Approaches · 6
Affine connection → In, Jean-Louis Koszul, Koszul, On, The, Weyl
related to Formal definition on the frame bundle · 6
Affine connection → An, FM, GL, In, Lie, Xξ
related to history · 6
Affine connection → Euclidean, However, On, Rn, Smooth, The
related to Curvature and torsion · 5
Affine connection → As, Cartan, Curvature, From, Maurer

Important terminology Word statistics

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Important terminology

affine connection tangent space vector bundle parallel manifold point transport curve frame connections principal gl aff rn cartan linear vectors

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Affine connectionis ageometric object on a smooth manifold which connects nearby tangent spaces0.90text
Affine connectionis a1-form η on P with values in aff0.90text
the point values of the vector fields Xξinstance ofRn which is horizontal in the sense that it vanishes on vertical vectors0.80text
Affine connectionrelated to Affine connections as Cartan connectionsAffine0.60section
Affine connectionrelated to Affine connections as Cartan connectionsCartan's0.60section
Affine connectionrelated to Affine connections as Cartan connectionsIn0.60section
Affine connectionrelated to Affine connections as Cartan connectionsIndeed0.60section
Affine connectionrelated to Affine connections as Cartan connectionsCartan0.60section
Affine connectionrelated to Affine connections as Cartan connectionsFrom0.60section
Affine connectionrelated to Affine connections as Cartan connectionsFM0.60section
Affine connectionrelated to Affine connections as Cartan connectionsHowever0.60section
Affine connectionrelated to ApproachesThe0.60section

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