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 y ∈ E the tangent mapping

is surjective, or, equivalently, its rank equals dim B.

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.

A triple (E, π, B) where E and B are differentiable manifolds and π: E → B is a surjective submersion, is called a fibered manifold.E is called the total space, B is called the base.

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