Fibered 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 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, or, equivalently, its rank equals dim $B$ .

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,^[2] 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 ^[3] under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.^[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,^[5] Whitney, Steenrod, Ehresmann,^[6]^[7]^[8] Serre,^[9] and others.

Formal definition

A triple $(E, π, B)$ where $E$ and $B$ are differentiable manifolds and $π : E \to B$ is a surjective submersion, is called a fibered manifold.^[10] 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 $(S 1 \times ℝ, π 1, S 1)$ and deleting two points in to two different fibers over the base manifold $S 1$ .The result is a new fibered manifold where all the fibers except two are connected.

Properties

Any surjective submersion $π : E \to B$ is open: for each open $V \subset E$ , the set $π (V) \subset B$ is open in $B$ .
Each fiber $π -1 (b) \subset E, b \in B$ is a closed embedded submanifold of $E$ of dimension $dim E - dim B$ .^[11]
A fibered manifold admits local sections: For each $y \in E$ there is an open neighborhood $U$ of $π (y)$ in $B$ and a smooth mapping $s : U \to E$ with $π \circ s = Id U$ and $s (π (y)) = y$ .
A surjection $π : E \to B$ is a fibered manifold if and only if there exists a local section $s : B \to E$ of $π$ (with $π \circ s = Id B$ ) passing through each $y \in E$ .^[12]

Fibered coordinates

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

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

where

{\begin{aligned}\psi &=(u^{1},\dots ,u^{n},y^{1},\dots ,y^{p-n}).\quad y_{0}\in V,\\\varphi &=(x^{1},\dots ,x^{n}),\quad \pi (y_{0})\in U.\end{aligned}}

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 projection onto the first $n$ coordinates. The chart $(U, φ)$ is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart $(V, ψ)$ are usually denoted by $ψ = (x i, y σ)$ where $i \in {1, ..., n}$ , $σ \in {1, ..., m}$ , $m = p - n$ the coordinates of the corresponding chart $U, φ)$ on $B$ are then denoted, with the obvious convention, by $φ = (x i)$ where $i \in {1, ..., n}$ .

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

Local trivialization and fiber bundles

Let $E \to B$ be a fibered manifold and $V$ any manifold. Then an open covering ${U α}$ of $B$ together with maps^[13]

\psi :\pi ^{-1}(U_{\alpha })\rightarrow U_{\alpha }\times V,

called trivialization maps, such that

\mathrm {pr} _{1}\circ \psi _{\alpha }=\pi ,\forall \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 $Ψ = {(U α, ψ α)}$ is then called a bundle atlas.

Notes

References

Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag
Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7
Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (1997). New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific. ISBN 981-02-1587-8.

Historical

Ehresmann, C. (1947a). "Sur la théorie des espaces fibrés". Coll. Top. alg. Paris (in French). C.N.R.S.: 3–15.
Ehresmann, C. (1947b). "Sur les espaces fibrés différentiables". C. R. Acad. Sci. Paris (in French). 224: 1611–1612.
Ehresmann, C. (1955). "Les prolongements d'un espace fibré différentiable". C. R. Acad. Sci. Paris (in French). 240: 1755–1757.
Feldbau, J. (1939). "Sur la classification des espaces fibrés". C. R. Acad. Sci. Paris (in French). 208: 1621–1623.
Seifert, H. (1932). "Topologie dreidimensionaler geschlossener Räume". Acta Math. (in French). 60: 147–238. doi:10.1007/bf02398271.
Serre, J.-P. (1951). "Homologie singulière des espaces fibrés. Applications". Ann. of Math. (in French). 54: 425–505. doi:10.2307/1969485.
Whitney, H. (1935). "Sphere spaces". Proc. Natl. Acad. Sci. USA. 21: 464–468. (open access)
Whitney, H. (1940). "On the theory of sphere bundles". Proc. Natl. Acad. Sci. USA. 26: 148–153. MR 0001338. (open access)

External links

McCleary, J. "A History of Manifolds and Fibre Spaces: Tortoises and Hares" (pdf).

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