In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion i.e. a surjective differentiable mapping such that at each point the tangent mapping is surjective (equivalently its rank equals dim B).
A triple where E and B are differentiable manifolds and is a surjective submersion, is called a fibered manifold. E is called the total space, B is called the base.
- 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 and deleting two points in to two different fibers over the base manifold .The result is a new fibered manifold where all the fibers except two are connected.
Let (resp. ) be an n-dimensional (resp. p-dimensional) manifold. A fibered manifold admits fiber charts. We say that a chart on is a fiber chart, or is adapted to the surjective submersion if there exists a chart on such that and
where with and with The above fiber chart condition may be equivalently expressed by
where is the first projection. The chart is then obviously unique. In view of the above property, the coordinates of a fiber chart are usually denoted by where the coordinates of the corresponding chart on are then denoted, with the obvious convention, by where
Any surjective submersion is open: for each open the set is open in .
A fibered manifold admits local sections: For each there is an open neighborhood of in and a smooth mapping with and .
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, 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  under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
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.
- Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, p. 11.
- Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, p. 47, ISBN 80-210-0165-8.
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- H. Whitney (1940). "On the theory of sphere bundles". Proc. Nat. Acad. Sci. USA 26: 148–153.
- J. Feldbau (1939). "Sur la classification des espaces fibrés". C. R. Acad. Sci. Paris 208: 1621–1623.
- C. Ehresmann (1947). "Sur la théorie des espaces fibrés". Coll. Top. alg. Paris. C.N.R.S.: 3–15.
- C. Ehresmann (1947). "Sur les espaces fibrés différentiables". C. R. Acad. Sci. Paris 224: 1611–1612.
- C. Ehresmann (1955). "Les prolongements d'un espace fibré différentiable". C. R. Acad. Sci. Paris 240: 1755–1757.
- J.-P. Serre (1951). "Homologie singulière des espaces fibrés. Applications". Ann. of Math. 54: 425–505. doi:10.2307/1969485.
- 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