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In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of , with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem.

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rdfs:label	Affine manifold (en)
rdfs:comment	In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of , with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem. (en)
dct:subject	Differential geometry Group theory Structures on manifolds Manifolds Affine geometry
Wikipage page ID	9996788 (xsd:integer)
Wikipage revision ID	1070142670 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press David Hilbert Homeomorphic Homomorphism Vector field Index of a subgroup Lie group (G,X)-manifold Differential geometry Group theory Structures on manifolds Connection (mathematics) Lorentz metric Ludwig Bieberbach Embedding Fundamental group Chern conjecture (affine geometry) Polycyclic group Affine connection Affine space Affine transformation Manifolds Hilbert's eighteenth problem Covering space Affine geometry Torsion tensor Web (differential geometry) Differentiable manifold Differential geometry Manifold Space group Cartan–Ambrose–Hicks theorem Euler class Chart (topology) Immersion (mathematics) Linear Parallelization (mathematics) Universal cover Universal covering Bieberbach group Flat connection Crystallographic group dbr:Auslander_conjecture dbr:Markus_conjecture
Link from a Wikipage to an external page	https://archive.org/details/affinedifferenti0000nomi https://archive.org/details/tensoranalysison00bish
sameAs	Affine manifold Affine manifold Affine manifold Affine manifold
dbp:wikiPageUsesTemplate	dbt:About-distinguish2 dbt:Citation dbt:Cite_book dbt:Reflist dbt:Sfn
has abstract	In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of , with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem. Equivalently, it is a manifold equipped with an atlas—called the affine structure—such that all transition functions between charts are affine transformations (that is, have constant Jacobian matrix); two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts. In each affine coordinate domain the coordinate vector fields form a parallelisation of that domain, so there is an associated connection on each domain. These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure. Note there is a link between linear connection (also called affine connection) and a web. (en)
prov:wasDerivedFrom	wikipedia-en:Affine_manifold?oldid=1070142670&ns=0
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foaf:isPrimaryTopicOf	wikipedia-en:Affine_manifold
is Link from a Wikipage to another Wikipage of	List of manifolds (G,X)-manifold Chern's conjecture (affine geometry) William Goldman (mathematician) Affine connection Kobayashi metric Homological mirror symmetry List of unsolved problems in mathematics Flat manifold Flat affine manifold
is Wikipage redirect of	Flat affine manifold
is foaf:primaryTopic of	wikipedia-en:Affine_manifold

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