About: Orientation of a vector bundle

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In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).

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rdfs:label	Orientation of a vector bundle (en)
rdfs:comment	In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n). (en)
dcterms:subject	Orientation (geometry) Analytic geometry Linear algebra
Wikipage page ID	44844703 (xsd:integer)
Wikipage revision ID	1073294835 (xsd:integer)
Link from a Wikipage to another Wikipage	Thom space General linear group Grassmannian Orientation sheaf Complex vector bundle Orientation (geometry) Analytic geometry Linear algebra Frame bundle Gysin sequence Special orthogonal group Orthogonal group Vector bundle Euler class Orientation bundle Orientation of a manifold Orientation of a vector space Determinant bundle Reduced cohomology Integration along the fiber Unit sphere bundle Standard orientation Thom isomorphism
sameAs	Orientation of a vector bundle Orientation of a vector bundle Orientation of a vector bundle
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Citation_needed dbt:Main dbt:Short_description
has abstract	In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) is fiberwise orientation-preserving, where Rn is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant. If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n). A vector bundle together with an orientation is called an oriented bundle. A vector bundle that can be given an orientation is called an orientable vector bundle. The basic invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a Gysin sequence. (en)
prov:wasDerivedFrom	wikipedia-en:Orientation_of_a_vector_bundle?oldid=1073294835&ns=0
page length (characters) of wiki page	4444 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Orientation_of_a_vector_bundle
is Link from a Wikipage to another Wikipage of	Integration along fibers Thom space Stiefel–Whitney class Orientation (vector space) Glossary of algebraic topology Glossary of differential geometry and topology Chern class Orientability Euler class Sphere bundle Orientable vector bundle Oriented bundle Oriented vector bundle
is Wikipage redirect of	Orientable vector bundle Oriented bundle Oriented vector bundle
is foaf:primaryTopic of	wikipedia-en:Orientation_of_a_vector_bundle

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