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 such that at each point yE 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[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,[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[edit]

A triple (E, π, B) where E and B are differentiable manifolds and π: EB is a surjective submersion, is called a fibered manifold.[10] 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 (S1 × ℝ, π1, S1) and deleting two points in to two different fibers over the base manifold S1.The result is a new fibered manifold where all the fibers except two are connected.

Properties[edit]

  • A surjection π : EB is a fibered manifold if and only if there exists a local section s : BE of π (with πs = IdB) passing through each yY.[11]
  • Any surjective submersion π: EB is open: for each open VE, the set π(V) ⊂ B is open in B.
  • Each fiber π−1(b) ⊂ E, bB is a closed embedded submanifold of E of dimension dim E − dim B.[12]
  • A fibered manifold admits local sections: For each yE there is an open neighborhood U of π(y) in B and a smooth mapping s: UE with πs = IdU and s(π(y)) = y.

Fibered coordinates[edit]

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: EB 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{align}\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{align}

The above fiber chart condition may be equivalently expressed by

\varphi\circ\pi = {\mathrm \pi_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, 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 = (xi, yσ) where i ∈ {1, ..., n}, σ ∈ {1, ..., m}, m = pn the coordinates of the corresponding chart U, φ) on B are then denoted, with the obvious convention, by φ = (xi) where i ∈ {1, ..., n}.

Conversely, if a surjection π: EB admits a fibered atlas, then π: EB is a fibered manifold.

Local trivialization and fiber bundles[edit]

Let EB be a fibered manifold and V any manifold. Then an open covering {Uα} 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 = {(Uα, ψα)} is then called a bundle atlas.

See also[edit]

Notes[edit]

References[edit]

  • 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 

Historical[edit]

  • 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. 

External links[edit]