Fiber Bundle

In mathematics, in particular topology, a fiber bundle is a continuous surjective map, π from a topological space E to another topological space B, satisfying a further condition making it locally of a particularly simple form. Putting it in intuitive terms, by locally here is meant locally on B; that is, if we imagine a small creature living on B and describing E, as mapped to B, only within a limited horizon on B, π has the description of a projection map inside a cartesian product. Introducing a further topological space F, we use the projection map from to B as the model. For example in the case of a vector bundle, F is a vector space over the real numbers. To qualify as a vector bundle, the matching conditions between local trivializable neighborhoods would have to be linear as well.

Saying it more formally, for any x in B, there is a neighborhood such that is homeomorphic to , in such a way that π carries over to the projection onto the first factor.

B is called the base space of the fiber bundle and for any , the preimage of x, is called the fiber at x and the map π is called the projection map.

A standard example is a Möbius strip as E, in which B can be taken as a circle and F a line segment. The 'twisting' in the band is only apparent globally, while locally the ribbon structure defines the topology.

Structure groups

Sometimes, there exists a topological group G of transformations of E, such that if ρ denotes the action,

for g in G and e in E. The condition states that every G-orbit lies within a single fiber.

In that case, G is called the structure group of the fiber bundle. To qualify as a G-bundle, the matching conditions between local trivializable neighborhoods would have to be intertwiners of G-actions as well.

If, in addition, G acts freely, transitively and continuously upon each fiber, then we call the fiber bundle a principal bundle. An example of a principal bundle that occurs naturally in geometry is the bundle of all bases for the tangent space to a manifold, with G the general linear group; restricting in Riemannian geometry to orthonormal bases, one would limit G to the orthogonal group. See vierbein for more details.

A section of a fiber bundle is a continuous map, such that , for x in B. Since bundles do not in general have sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology.

One of the primary applications of fiber bundles is in gauge theory.

See also Fibration.

Modern Translations: Fiber Bundle

Anagrams: Fiber Bundle

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "b-b-d-e-e-f-i-l-n-r-u"
	-2 letters: unbriefed.
	-3 letters: befriend, belfried, bluebird, fribbled, linebred, nubblier, rendible, unbelief, unbridle, underlie, unedible, unrifled.
	-4 letters: berline, blender, blinder, bluefin, blunder, blurbed, briefed, brindle, builder, bundler, burbled, debrief, defiler, definer, dibbler, direful, dribble, febrile, feruled, fibered, fielder, flinder, flubbed, flubber, fribble, inbreed, needful, nibbled, nibbler, nubbier, reblend, rebuild, redline, refiled, refined, relined, rubbled, unfired.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: Fiber Bundle

Hexadecimal (or equivalents, 770AD-1900s) (references)

46 69 62 65 72 42 75 6E 64 6C 65

Leonardo da Vinci (1452-1519; backwards) (references)

Binary Code (1918-1938, probably earlier) (references)

01000110 01101001 01100010 01100101 01110010 00100000 01000010 01110101 01101110 01100100 01101100 01100101

HTML Code (1990) (references)

&#70 &#105 &#98 &#101 &#114 &#32 &#66 &#117 &#110 &#100 &#108 &#101

ISO 10646 (1991-1993) (references)

0046 0069 0062 0065 0072 0042 0075 006E 0064 006C 0065

Encryption (beginner's substitution cypher): (references)

Language		Translations for "fiber bundle"; alternative meanings/domain in parentheses.
Japanese Kanji		繊維束 (fascicle), 線維束 (fascicle). (various references)

Japanese Katakana		せにそく (fascicle). (various references)

Pig Latin		iberfay undlebay
Source: compiled by the editor from various translation references.

Fiber Bundle

Definition: Fiber Bundle

Fiber Bundle

Noun

Synonyms: Fiber Bundle

Specialty Definition: Fiber bundle

Structure groups

Crosswords: Fiber Bundle

Modern Translations: Fiber Bundle

Japanese Kanji

Japanese Katakana

Pig Latin

Anagrams: Fiber Bundle

Scrabble® Enable2K-Verified Anagrams

Alternative Orthography: Fiber Bundle

46 69 62 65 72 42 75 6E 64 6C 65

01000110 01101001 01100010 01100101 01110010 00100000 01000010 01110101 01101110 01100100 01101100 01100101

&#70 &#105 &#98 &#101 &#114 &#32 &#66 &#117 &#110 &#100 &#108 &#101

0046 0069 0062 0065 0072 0042 0075 006E 0064 006C 0065

40756871842368780707871

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