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Statements

Subject Item: dbr:Integration_along_fibers
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Thom_space
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Stiefel–Whitney_class
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Orientation_(vector_space)
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Orientation_of_a_vector_bundle
rdfs:label: Orientation of a vector bundle
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).
dcterms:subject: dbc:Orientation_(geometry) dbc:Analytic_geometry dbc:Linear_algebra
dbo:wikiPageID: 44844703
dbo:wikiPageRevisionID: 1073294835
dbo:wikiPageWikiLink: dbr:Vector_bundle dbr:Orthogonal_group dbr:Reduced_cohomology dbr:Gysin_sequence dbc:Analytic_geometry dbr:Orientation_bundle dbc:Linear_algebra dbr:Special_orthogonal_group dbr:Thom_space dbr:Grassmannian dbr:Complex_vector_bundle dbc:Orientation_(geometry) dbr:Frame_bundle dbr:Orientation_sheaf dbr:Orientation_of_a_manifold dbr:General_linear_group dbr:Orientation_of_a_vector_space dbr:Thom_isomorphism dbr:Standard_orientation dbr:Determinant_bundle dbr:Unit_sphere_bundle dbr:Euler_class dbr:Integration_along_the_fiber
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dbo: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.
prov:wasDerivedFrom: wikipedia-en:Orientation_of_a_vector_bundle?oldid=1073294835&ns=0
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foaf:isPrimaryTopicOf: wikipedia-en:Orientation_of_a_vector_bundle

Subject Item: dbr:Glossary_of_algebraic_topology
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Glossary_of_differential_geometry_and_topology
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Chern_class
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Orientability
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Euler_class
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Sphere_bundle
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Orientable_vector_bundle
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle
dbo:wikiPageRedirects: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Oriented_bundle
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle
dbo:wikiPageRedirects: dbr:Orientation_of_a_vector_bundle

Subject Item: dbr:Oriented_vector_bundle
dbo:wikiPageWikiLink: dbr:Orientation_of_a_vector_bundle
dbo:wikiPageRedirects: dbr:Orientation_of_a_vector_bundle

Subject Item: wikipedia-en:Orientation_of_a_vector_bundle
foaf:primaryTopic: dbr:Orientation_of_a_vector_bundle