# Fibered manifold

(Redirected from Fibred manifold)

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

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

• 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

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

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.

## Notes

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.