释义 |
- History
- Formal definition
- Examples
- Properties
- Fibered coordinates
- Local trivialization and fiber bundles
- See also
- Notes
- References Historical
- External links
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion[1] i.e. a surjective differentiable mapping such that at each point {{math|y ∈ E}} the tangent mapping is surjective, or, equivalently, its rank equals dim {{math|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 {{math|(E, π, B)}} where {{math|E}} and {{math|B}} are differentiable manifolds and {{math|π: E → 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 {{math|(S1 × ℝ, π1, S1)}} and deleting two points in two different fibers over the base manifold {{math|S1}}.The result is a new fibered manifold where all the fibers except two are connected.
Properties - Any surjective submersion {{math|π: E → B}} is open: for each open {{math|V ⊂ E}}, the set {{math|π(V) ⊂ B}} is open in {{math|B}}.
- Each fiber {{math|π−1(b) ⊂ E, b ∈ B}} is a closed embedded submanifold of {{mvar|E}} of dimension {{math|dim E − dim B}}.[11]
- A fibered manifold admits local sections: For each {{math|y ∈ E}} there is an open neighborhood {{math|U}} of {{math|π(y)}} in {{math|B}} and a smooth mapping {{math|s: U → E}} with {{math|1=π ∘ s = IdU}} and {{math|s(π(y)) {{=}} y}}.
- A surjection {{math|π : E → B}} is a fibered manifold if and only if there exists a local section {{math|s : B → E}} of {{mvar|π}} (with {{math|π ∘ s {{=}} IdB}}) passing through each {{math|y ∈ E}}.[12]
Fibered coordinates Let {{math|B}} (resp. {{math|E}}) be an {{math|n}}-dimensional (resp. {{math|p}}-dimensional) manifold. A fibered manifold {{math|(E, π, B)}} admits fiber charts. We say that a chart {{math|(V, ψ)}} on {{math|E}} is a fiber chart, or is adapted to the surjective submersion {{math|π: E → B}} if there exists a chart {{math|(U, φ)}} on {{math|B}} such that {{math|1=U = π(V)}} and where The above fiber chart condition may be equivalently expressed by where is the projection onto the first {{math|n}} coordinates. The chart {{math|(U, φ)}} is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart {{math|(V, ψ)}} are usually denoted by {{math|ψ {{=}} (xi, yσ)}} where {{math|i ∈ {1, ..., n}}}, {{math|σ ∈ {1, ..., m}}}, {{math|1=m = p − n}} the coordinates of the corresponding chart {{math|U, φ)}} on {{math|B}} are then denoted, with the obvious convention, by {{math|1=φ = (xi)}} where {{math|i ∈ {1, ..., n}}}. Conversely, if a surjection {{math|π: E → B}} admits a fibered atlas, then {{math|π: E → B}} is a fibered manifold. Local trivialization and fiber bundles Let {{math|E → B}} be a fibered manifold and {{math|V}} any manifold. Then an open covering {{math|{Uα}}} of {{math|B}} together with maps[13] called trivialization maps, such that is a local trivialization with respect to {{math|V}}. A fibered manifold together with a manifold {{math|V}} is a fiber bundle with typical fiber (or just fiber) {{math|V}} if it admits a local trivialization with respect to {{mvar|V}}. The atlas {{math|Ψ {{=}} {(Uα, ψα)}}} is then called a bundle atlas. See also - Covering space
- Fiber bundle
- Fibration
- Quasi-fibration
- Natural bundle
- Seifert fiber space
- Connection (fibred manifold)
- Algebraic fiber space
Notes 1. ^{{harvnb|Kolář|1993|p=11}} 2. ^{{harvnb|Seifert|1932}} 3. ^{{harvnb|Whitney|1935}} 4. ^{{harvnb|Whitney|1940}} 5. ^{{harvnb|Feldbau|1939}} 6. ^{{harvnb|Ehresman|1947a}} 7. ^{{harvnb|Ehresman|1947b}} 8. ^{{harvnb|Ehresman|1955}} 9. ^{{harvnb|Serre|1951}} 10. ^{{harvnb|Krupka|Janyška|1990|p=47}} 11. ^{{harvnb|Giachetta|Mangiarotti|Sardanashvily|1997|p=11}} 12. ^{{harvnb|Giachetta|Mangiarotti|Sardanashvily|1997|p=15}} 13. ^{{harvnb|Giachetta|Mangiarotti|Sardanashvily|1997|p=13}}
References - {{citation|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag}}
- {{citation|last1 = Krupka|first1=Demeter|last2=Janyška|first2=Josef|title=Lectures on differential invariants|year = 1990|publisher = Univerzita J. E. Purkyně V Brně|isbn=80-210-0165-8}}
- {{citation|last1 = Saunders|first1=D.J.|title=The geometry of jet bundles|year = 1989|publisher = Cambridge University Press|isbn=0-521-36948-7}}
- {{cite book|ref=harv|last1=Giachetta|first1=G.|last2=Mangiarotti|first2=L.|last3=Sardanashvily|first3=G.|authorlink3=Gennadi Sardanashvily|title=New Lagrangian and Hamiltonian Methods in Field Theory|publisher=World Scientific|year=1997|isbn=981-02-1587-8}}
Historical - {{cite journal|ref=harv|title=Sur la théorie des espaces fibrés|first=C.|last=Ehresmann|authorlink=Charles Ehresmann|journal=Coll. Top. alg. Paris|volume=C.N.R.S.|year=1947a|pages=3–15|language=French}}
- {{cite journal|ref=harv|title=Sur les espaces fibrés différentiables|first=C.|last=Ehresmann|journal=C. R. Acad. Sci. Paris|volume=224|year=1947b|pages=1611–1612|language=French}}
- {{cite journal|ref=harv|title=Les prolongements d'un espace fibré différentiable|first=C.|last=Ehresmann|journal=C. R. Acad. Sci. Paris|volume=240|year=1955|pages=1755–1757|language=French}}
- {{cite journal|ref=harv|title=Sur la classification des espaces fibrés|first=J.|last=Feldbau|authorlink=Jacques Feldbau|journal=C. R. Acad. Sci. Paris|volume=208|year=1939|pages=1621–1623|language=French}}
- {{cite journal|ref=harv|title=Topologie dreidimensionaler geschlossener Räume|first=H.|last=Seifert|authorlink=Herbert Seifert|journal=Acta Math.|volume=60|year=1932|pages=147–238|doi=10.1007/bf02398271|language=French}}
- {{cite journal|ref=harv|title=Homologie singulière des espaces fibrés. Applications|first=J.-P.|last=Serre|authorlink=Jean-Pierre Serre|journal=Ann. of Math.|volume=54|year=1951|pages=425–505|doi=10.2307/1969485|language=French}}
- {{cite journal|ref=harv|title=Sphere spaces|first=H.|last=Whitney|authorlink=Hassler Whitney|journal=Proc. Natl. Acad. Sci. USA|volume=21|year=1935|pages=464–468|doi=10.1073/pnas.21.7.464|url=http://www.pnas.org/content/21/7.toc}} {{open access}}
- {{cite journal|ref=harv|title=On the theory of sphere bundles|first=H.|last=Whitney|journal=Proc. Natl. Acad. Sci. USA|volume=26|year=1940|pages=148–153|mr=0001338|url=http://www.pnas.org/content/26/2.toc|doi=10.1073/pnas.26.2.148}} {{open access}}
External links - {{cite web|title=A History of Manifolds and Fibre Spaces: Tortoises and Hares|url=http://pages.vassar.edu/mccleary/files/2011/04/history.fibre_.spaces.pdf|format=pdf|last=McCleary|first=J.}}
3 : Differential geometry|Manifolds|Fiber bundles |