Fibered manifold

Summary

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion

that is, a surjective differentiable mapping such that at each point the tangent mapping
is surjective, or, equivalently, its rank equals [1]

History edit

In topology, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1932, but his definitions are limited to a very special case.[2] 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   was not part of the structure, but derived from it as a quotient space of   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.[3][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, Whitney, Steenrod, Ehresmann, Serre, and others.[5][6][7][8][9]

Formal definition edit

A triple   where   and   are differentiable manifolds and   is a surjective submersion, is called a fibered manifold.[10]   is called the total space,   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 need not 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 two different fibers over the base manifold   The result is a new fibered manifold where all the fibers except two are connected.

Properties edit

  • Any surjective submersion   is open: for each open   the set   is open in  
  • Each fiber   is a closed embedded submanifold of   of dimension  [11]
  • A fibered manifold admits local sections: For each   there is an open neighborhood   of   in   and a smooth mapping   with   and  
  • A surjection   is a fibered manifold if and only if there exists a local section   of   (with  ) passing through each  [12]

Fibered coordinates edit

Let   (resp.  ) be an  -dimensional (resp.  -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
 

The above fiber chart condition may be equivalently expressed by

 
where
 
is the projection onto the first   coordinates. The chart   is then obviously unique. In view of the above property, the fibered 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  

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

Local trivialization and fiber bundles edit

Let   be a fibered manifold and   any manifold. Then an open covering   of   together with maps

 
called trivialization maps, such that
 
is a local trivialization with respect to  [13]

A fibered manifold together with a manifold   is a fiber bundle with typical fiber (or just fiber)   if it admits a local trivialization with respect to   The atlas   is then called a bundle atlas.

See also edit

Notes edit

References edit

  • Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on March 30, 2017, retrieved June 15, 2011
  • 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 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.
  • 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. JSTOR 1969485.
  • Whitney, H. (1935). "Sphere spaces". Proc. Natl. Acad. Sci. USA. 21 (7): 464–468. Bibcode:1935PNAS...21..464W. doi:10.1073/pnas.21.7.464. PMC 1076627. PMID 16588001.  
  • Whitney, H. (1940). "On the theory of sphere bundles". Proc. Natl. Acad. Sci. USA. 26 (2): 148–153. Bibcode:1940PNAS...26..148W. doi:10.1073/pnas.26.2.148. MR 0001338. PMC 1078023. PMID 16588328.  

External links edit

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