Fibered manifold

 ( Differential Geometry ) 

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion $$\backslash pi\; :\; E\; \backslash to\; B\backslash ,$$ that is, a surjective differentiable mapping such that at each point $y\; \backslash in\; E$ the tangent mapping $$T\_y\; \backslash pi\; :\; T\_\{y\}\; E\; \backslash to\; T\_\{\backslash pi(y)\}B$$ is surjective, or, equivalently, its rank equals $\backslash dim\; B.$

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History In topology, the words fiber ( Faser in German) and fiber space ( gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1932, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space $E$ was not part of the structure, but derived from it as a quotient space of $E.$ The first definition of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.

The theory of fibered spaces, of which , , topological and fibered manifolds are a special case, is attributed to Herbert Seifert, Heinz Hopf, Jacques Feldbau, Hassler Whitney, Norman Steenrod, Ehresmann, Serre, and others.

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Formal definition A triple $(E,\; \backslash pi,\; B)$ where $E$ and $B$ are differentiable manifolds and $\backslash pi\; :\; E\; \backslash to\; B$ is a surjective submersion, is called a fibered manifold. $E$ is called the total space, $B$ is called the base.

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Examples

• Every differentiable fiber bundle is a fibered manifold.
• Every differentiable covering space is a fibered manifold with discrete fiber.
• In general, a fibered manifold need not be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by restricting projection to total space of any vector bundle over smooth manifold with removed base space embedded by global section, finitely many points or any closed submanifold not containing a fiber in general.

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Properties

• Any surjective submersion $\backslash pi\; :\; E\; \backslash to\; B$ is open: for each open $V\; \backslash subseteq\; E,$ the set $\backslash pi(V)\; \backslash subseteq\; B$ is open in $B.$
• Each fiber $\backslash pi^\{-1\}(b)\; \backslash subseteq\; E,\; b\; \backslash in\; B$ is a closed embedded submanifold of $E$ of dimension $\backslash dim\; E\; -\; \backslash dim\; B.$
• A fibered manifold admits local sections: For each $y\; \backslash in\; E$ there is an open neighborhood $U$ of $\backslash pi(y)$ in $B$ and a smooth mapping $s\; :\; U\; \backslash to\; E$ with $\backslash pi\; \backslash circ\; s\; =\; \backslash operatorname\{Id\}\_U$ and $s(\backslash pi(y))\; =\; y.$
• A surjection $\backslash pi\; :\; E\; \backslash to\; B$ is a fibered manifold if and only if there exists a local section $s\; :\; B\; \backslash to\; E$ of $\backslash pi$ (with $\backslash pi\; \backslash circ\; s\; =\; \backslash operatorname\{Id\}\_B$) passing through each $y\; \backslash in\; E.$

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Fibered coordinates Let $B$ (resp. $E$) be an $n$-dimensional (resp. $p$-dimensional) manifold. A fibered manifold $(E,\; \backslash pi,\; B)$ admits fiber charts. We say that a chart $(V,\; \backslash psi)$ on $E$ is a fiber chart, or is adapted to the surjective submersion $\backslash pi\; :\; E\; \backslash to\; B$ if there exists a chart $(U,\; \backslash varphi)$ on $B$ such that $U\; =\; \backslash pi(V)$ and $$u^1\; =\; x^1\backslash circ\; \backslash pi,\backslash ,u^2\; =\; x^2\backslash circ\; \backslash pi,\backslash ,\backslash dots,\backslash ,u^n\; =\; x^n\backslash circ\; \backslash pi\backslash ,\; ,$$ where

The above fiber chart condition may be equivalently expressed by $$\backslash varphi\backslash circ\backslash pi\; =\; \backslash mathrm\{pr\}\_1\backslash circ\backslash psi,$$ where $$\{\backslash mathrm\; \{pr\}\_1\}\; :\; \{\backslash R^n\}\backslash times\{\backslash R^\{p-n\}\}\; \backslash to\; \{\backslash R^n\}\backslash ,$$ is the projection onto the first $n$ coordinates. The chart $(U,\; \backslash varphi)$ is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart $(V,\; \backslash psi)$ are usually denoted by $\backslash psi\; =\; \backslash left(x^i,\; y^\{\backslash sigma\}\backslash right)$ where $i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \},$ $\backslash sigma\; \backslash in\; \backslash \{1,\; \backslash ldots,\; m\backslash \},$ $m\; =\; p\; -\; n$ the coordinates of the corresponding chart $(U,\; \backslash varphi)$ on $B$ are then denoted, with the obvious convention, by $\backslash varphi\; =\; \backslash left(x\_i\backslash right)$ where $i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}.$

Conversely, if a surjection $\backslash pi\; :\; E\; \backslash to\; B$ admits a fibered atlas, then $\backslash pi\; :\; E\; \backslash to\; B$ is a fibered manifold.

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Local trivialization and fiber bundles Let $E\; \backslash to\; B$ be a fibered manifold and $V$ any manifold. Then an open covering $\backslash left\backslash \{U\_\{\backslash alpha\}\backslash right\backslash \}$ of $B$ together with maps $$\backslash psi\; :\; \backslash pi^\{-1\}\backslash left(U\_\backslash alpha\backslash right)\; \backslash to\; U\_\backslash alpha\; \backslash times\; V,$$ called trivialization maps, such that $$\backslash mathrm\{pr\}\_1\; \backslash circ\; \backslash psi\_\backslash alpha\; =\; \backslash pi,\; \backslash text\{\; for\; all\; \}\; \backslash alpha$$ is a local trivialization with respect to $V.$

A fibered manifold together with a manifold $V$ is a fiber bundle with typical fiber (or just fiber) $V$ if it admits a local trivialization with respect to $V.$ The atlas $\backslash Psi\; =\; \backslash left\backslash \{\backslash left(U\_\{\backslash alpha\},\; \backslash psi\_\{\backslash alpha\}\backslash right)\backslash right\backslash \}$ is then called a bundle atlas.

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See also

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Notes

• (1997). 9789810215873, World Scientific. ISBN 9789810215873

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Historical

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External links

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