Fibered manifold

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In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion [1] \pi \colon E \to B\, , i.e. a surjective differentiable mapping \pi \colon E \to B\, such that at each point y\in E the tangent mapping T_y\pi \colon T_{y}E \to T_{\pi(y)}B is surjective (equivalently its rank equals dim B).

Formal definition[edit]

A triple (E,\pi,B)\, , where E and B are differentiable manifolds and \pi \colon E \to B\, is a surjective submersion, is called a fibered manifold.[2] E is called the total space, B is called the base.

Examples[edit]

  • Every differentiable fiber bundle is a fibered manifold.
  • Every differentiable covering space is a fibered manifold with discrete fiber.
  • In general, a fibered manifold needs not to be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle ({\mathbb {S}^1}\times {\mathbb {R}^1},{\mathrm {pr}_1},{\mathbb {S}^1})\, , and deleting two points in to two different fibers over the base manifold {\mathbb {S}^1}\,.The result is a new fibered manifold where all the fibers except two are connected.

Properties[edit]

Let B (resp. E) be an n-dimensional (resp. p-dimensional) manifold. A fibered manifold (E,\pi,B)\, admits fiber charts. We say that a chart (V,\psi)\, on E\, is a fiber chart, or is adapted to the surjective submersion \pi \colon E \to B\, , if there exists a chart (U,\varphi)\, on B\, such that U=\pi (V)\, and

u^1=x^1\circ \pi,\,u^2=x^2\circ \pi,\,\dots,\,u^n=x^n\circ \pi\, ,

where \psi=(u^1,\dots,u^n,y^1,\dots,y^{p-n})\, with y_{0}\in V\, and \varphi=(x^1,\dots,x^n)\, with \pi(y_{0})\in U\, . The above fiber chart condition may be equivalently expressed by

\varphi\circ\pi={\mathrm {pr}_1}\circ\psi\, ,

where {\mathrm {pr}_1} \colon {\mathbb R^n}\times{\mathbb R^{p-n}} \to {\mathbb R^n}\, is the first projection. The chart (U,\varphi)\, is then obviously unique. In view of the above property, the coordinates of a fiber chart (V,\psi)\, are usually denoted by \psi = (x^i,y^{\sigma})\, , where i\in \{1,\dots,n\}\, , \sigma\in \{1,\dots,m\}\, , m=p-n\, ; the coordinates of the corresponding chart (U,\varphi)\, on B\, are then denoted, with the obvious convention, by \varphi = (x^i)\, , where i\in \{1,\dots,n\}\, .

Any surjective submersion \pi \colon E \to B\, is open: for each open V\subset E\, the set \pi(V)\subset B\, is open in B\,.

A fibered manifold admits local sections: For each y\in E\, there is an open neighborhood U\, of \pi(y)\, in B\, and a smooth mapping s\colon U\to E\, with \pi\circ s={\mathrm {Id}_U}\, and s(\pi(y))=y\,.

History[edit]

In topology, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by Seifert in 1932,[3] 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 [4] under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.[5]

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,[6] Whitney, Steenrod, Ehresmann,[7][8][9] Serre,[10] and others.

See also[edit]

Notes[edit]

  1. ^ Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag . (page 11)
  2. ^ Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8 . (page 47)
  3. ^ H. Seifert (1932). "Topologie dreidimensionaler geschlossener Räume". Acta Math. 60: 147–238. 
  4. ^ H. Whitney (1935). "Sphere spaces". Proc. Nat. Acad. Sci. USA 21: 464–468. 
  5. ^ H. Whitney (1940). "On the theory of sphere bundles". Proc. Nat. Acad. Sci. USA 26: 148–153. 
  6. ^ J. Feldbau (1939). "Sur la classification des espaces fibrés". C. R. Acad. Sci. Paris 208: 1621–1623. 
  7. ^ C. Ehresmann (1947). "Sur la théorie des espaces fibrés". Coll. Top. alg. Paris. C.N.R.S.: 3–15. 
  8. ^ C. Ehresmann (1947). "Sur les espaces fibrés différentiables". C. R. Acad. Sci. Paris 224: 1611–1612. 
  9. ^ C. Ehresmann (1955). "Les prolongements d'un espace fibré différentiable". C. R. Acad. Sci. Paris 240: 1755–1757. 
  10. ^ J.-P. Serre (1951). "Homologie singulière des espaces fibrés. Applications". Ann. of Math. 54: 425–505. 

References[edit]