Concept in differential geometry
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion
![{\displaystyle \pi :E\to B\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ef33996a94370a618290de8cb7131acea932e5)
that is, a surjective differentiable mapping such that at each point
the tangent mapping
![{\displaystyle T_{y}\pi :T_{y}E\to T_{\pi (y)}B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb3bfa23fce2bdb2b3655b457bf47c48d561b04)
is surjective, or, equivalently, its rank equals
[1]
History
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]
Fibered coordinates
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
![{\displaystyle u^{1}=x^{1}\circ \pi ,\,u^{2}=x^{2}\circ \pi ,\,\dots ,\,u^{n}=x^{n}\circ \pi \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/166c78654b45bd21f17b8f5a845e6a1ac9ae2d90)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bef88a7c94dde98fc0770327b22dec9fbd912558)
The above fiber chart condition may be equivalently expressed by
![{\displaystyle \varphi \circ \pi =\mathrm {pr} _{1}\circ \psi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a50f493d593bd32f63e286280bcb542aeb860c1)
where
![{\displaystyle {\mathrm {pr} _{1)):{\mathbb {R} ^{n))\times {\mathbb {R} ^{p-n))\to {\mathbb {R} ^{n))\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5025ef2859bba642487f6e80e0e5c39ae260214e)
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.