Fibered manifold

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, or, equivalently, its rank equals dim .
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, 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.
Formal definition.
A triple where and are differentiable manifolds and is a surjective submersion, is called a fibered manifold. "E" is called the total space, "B" is called the base.
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
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.
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 .
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.