Fibered manifold

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 ${\displaystyle E}$ was not part of the structure, but derived from it as a quotient space of ${\displaystyle E.}$ 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.[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]

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

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

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

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

where

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

The above fiber chart condition may be equivalently expressed by

$${\displaystyle \varphi \circ \pi =\mathrm {pr} _{1}\circ \psi ,}$$

where

$${\displaystyle {\mathrm {pr} _{1}}:{\mathbb {R} ^{n}}\times {\mathbb {R} ^{p-n}}\to {\mathbb {R} ^{n}}\,}$$

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

Conversely, if a surjection ${\displaystyle \pi :E\to B}$ admits a fibered atlas, then ${\displaystyle \pi :E\to B}$ is a fibered manifold.

Let ${\displaystyle E\to B}$ be a fibered manifold and ${\displaystyle V}$ any manifold. Then an open covering ${\displaystyle \left\{U_{\alpha }\right\}}$ of ${\displaystyle B}$ together with maps

$${\displaystyle \psi$$ :\pi ^{-1}\left(U_{\alpha }\right)\to U_{\alpha }\times V,}

called trivialization maps, such that

$${\displaystyle \mathrm {pr} _{1}\circ \psi _{\alpha }=\pi ,{\text{ for all }}\alpha }$$

is a local trivialization with respect to ${\displaystyle V.}$[13]

A fibered manifold together with a manifold ${\displaystyle V}$ is a fiber bundle with typical fiber (or just fiber) ${\displaystyle V}$ if it admits a local trivialization with respect to ${\displaystyle V.}$ The atlas ${\displaystyle \Psi =\left\{\left(U_{\alpha },\psi _{\alpha }\right)\right\}}$ is then called a bundle atlas.

• Algebraic fiber space
• Connection (fibred manifold) – Operation on fibered manifolds
• Covering space – Type of continuous map in topology
• Fiber bundle – Continuous surjection satisfying a local triviality condition
• Fibration – Concept in algebraic topology
• Natural bundle
• Quasi-fibration – Concept from mathematics
• Seifert fiber space – Topological space

• 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.

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