I-bundle (original) (raw)
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial. Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle . That surface has three I-bundles: the trivial bundle and two twisted bundles.
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| dbo:abstract | In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial. Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product , and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold. Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle . That surface has three I-bundles: the trivial bundle and two twisted bundles. Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces. These observations are simple well known facts on elementary 3-manifolds. Line bundles are both I-bundles and vector bundles of rank one. When considering I-bundles, one is interested mostly in their topological properties and not their possible vector properties, as one might be for line bundles. (en) |
| dbo:thumbnail | wiki-commons:Special:FilePath/MobiusF.png?width=300 |
| dbo:wikiPageExternalLink | https://web.archive.org/web/20120722220119/https:/www.math.lsu.edu/~kasten/LSUTalk.pdf |
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| rdfs:comment | In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial. Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle . That surface has three I-bundles: the trivial bundle and two twisted bundles. (en) |
| rdfs:label | I-bundle (en) |
| owl:sameAs | freebase:I-bundle yago-res:I-bundle wikidata:I-bundle https://global.dbpedia.org/id/4nQjg |
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| foaf:depiction | wiki-commons:Special:FilePath/Hopf_band_wikipedia.png wiki-commons:Special:FilePath/MobiusF.png wiki-commons:Special:FilePath/MxS1.png |
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