Английская Википедия:Fibered manifold
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In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion <math display=block>\pi : E \to B\,</math> that is, a surjective differentiable mapping such that at each point <math>y \in U</math> the tangent mapping <math display=block>T_y \pi : T_{y} E \to T_{\pi(y)}B</math> is surjective, or, equivalently, its rank equals <math>\dim B.</math>[1]
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.[2] 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 <math>E</math> was not part of the structure, but derived from it as a quotient space of <math>E.</math> 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.[3][4]
The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Hopf, Feldbau, Whitney, Steenrod, Ehresmann, Serre, and others.[5][6][7][8][9]
Formal definition
A triple <math>(E, \pi, B)</math> where <math>E</math> and <math>B</math> are differentiable manifolds and <math>\pi : E \to B</math> is a surjective submersion, is called a fibered manifold.[10] <math>E</math> is called the total space, <math>B</math> is called the base.
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 taking the trivial bundle <math>\left(S^1 \times \R, \pi_1, S^1\right)</math> and deleting two points in two different fibers over the base manifold <math>S^1.</math> The result is a new fibered manifold where all the fibers except two are connected.
Properties
- Any surjective submersion <math>\pi : E \to B</math> is open: for each open <math>V \subseteq E,</math> the set <math>\pi(V) \subseteq B</math> is open in <math>B.</math>
- Each fiber <math>\pi^{-1}(b) \subseteq E, b \in B</math> is a closed embedded submanifold of <math>E</math> of dimension <math>\dim E - \dim B.</math>[11]
- A fibered manifold admits local sections: For each <math>y \in E</math> there is an open neighborhood <math>U</math> of <math>\pi(y)</math> in <math>B</math> and a smooth mapping <math>s : U \to E</math> with <math>\pi \circ s = \operatorname{Id}_U</math> and <math>s(\pi(y)) = y.</math>
- A surjection <math>\pi : E \to B</math> is a fibered manifold if and only if there exists a local section <math>s : B \to E</math> of <math>\pi</math> (with <math>\pi \circ s = \operatorname{Id}_B</math>) passing through each <math>y \in E.</math>[12]
Fibered coordinates
Let <math>B</math> (resp. <math>E</math>) be an <math>n</math>-dimensional (resp. <math>p</math>-dimensional) manifold. A fibered manifold <math>(E, \pi, B)</math> admits fiber charts. We say that a chart <math>(V, \psi)</math> on <math>E</math> is a fiber chart, or is adapted to the surjective submersion <math>\pi : E \to B</math> if there exists a chart <math>(U, \varphi)</math> on <math>B</math> such that <math>U = \pi(V)</math> and <math display=block>u^1 = x^1\circ \pi,\,u^2 = x^2\circ \pi,\,\dots,\,u^n = x^n\circ \pi\, ,</math> where <math display=block>\begin{align}\psi &= \left(u^1,\dots,u^n,y^1,\dots,y^{p-n}\right). \quad y_{0}\in V,\\ \varphi &= \left(x^1,\dots,x^n\right), \quad \pi\left(y_0\right)\in U.\end{align}</math>
The above fiber chart condition may be equivalently expressed by <math display=block>\varphi\circ\pi = \mathrm{pr}_1\circ\psi,</math> where <math display=block>{\mathrm {pr}_1} : {\R^n}\times{\R^{p-n}} \to {\R^n}\,</math> is the projection onto the first <math>n</math> coordinates. The chart <math>(U, \varphi)</math> is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart <math>(V, \psi)</math> are usually denoted by <math>\psi = \left(x^i, y^{\sigma}\right)</math> where <math>i \in \{1, \ldots, n\},</math> <math>\sigma \in \{1, \ldots, m\},</math> <math>m = p - n</math> the coordinates of the corresponding chart <math>(U, \varphi)</math> on <math>B</math> are then denoted, with the obvious convention, by <math>\varphi = \left(x_i\right)</math> where <math>i \in \{1, \ldots, n\}.</math>
Conversely, if a surjection <math>\pi : E \to B</math> admits a fibered atlas, then <math>\pi : E \to B</math> is a fibered manifold.
Local trivialization and fiber bundles
Let <math>E \to B</math> be a fibered manifold and <math>V</math> any manifold. Then an open covering <math>\left\{U_{\alpha}\right\}</math> of <math>B</math> together with maps <math display=block>\psi : \pi^{-1}\left(U_\alpha\right) \to U_\alpha \times V,</math> called trivialization maps, such that <math display=block>\mathrm{pr}_1 \circ \psi_\alpha = \pi, \text{ for all } \alpha</math> is a local trivialization with respect to <math>V.</math>[13]
A fibered manifold together with a manifold <math>V</math> is a fiber bundle with typical fiber (or just fiber) <math>V</math> if it admits a local trivialization with respect to <math>V.</math> The atlas <math>\Psi = \left\{\left(U_{\alpha}, \psi_{\alpha}\right)\right\}</math> is then called a bundle atlas.
See also
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Notes
References
Historical
- Шаблон:Cite journal
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- Шаблон:Cite journal Шаблон:Open access
- Шаблон:Cite journal Шаблон:Open access
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