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 y \in U the tangent mapping is surjective, or, equivalently, its rank equals \dim B.

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. 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 (E, \pi, B) where E and B are differentiable manifolds and is a surjective submersion, is called a fibered manifold. E is called the total space, B is called the base.

Examples

Properties

Fibered coordinates

Let B (resp. E) be an n-dimensional (resp. p-dimensional) manifold. A fibered manifold (E, \pi, B) admits fiber charts. We say that a chart (V, \psi) on E is a fiber chart, or is adapted to the surjective submersion if there exists a chart on B such that U = \pi(V) and where The above fiber chart condition may be equivalently expressed by where is the projection onto the first n coordinates. The chart is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart (V, \psi) are usually denoted by where m = p - n the coordinates of the corresponding chart on B 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 E \to B be a fibered manifold and V any manifold. Then an open covering of B together with maps called trivialization maps, such that is a local trivialization with respect to V. A fibered manifold together with a manifold V is a fiber bundle with typical fiber (or just fiber) V if it admits a local trivialization with respect to V. The atlas is then called a bundle atlas.

Historical