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 the tangent mapping is surjective, or, equivalently, its rank equals 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 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. 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. is called the total space, is called the base. Examples Every differentiable fiber bundle is a fibered manifold. Every differentiable covering space is a fibered manifold with discrete fiber. In general, a fibered manifold needs not to be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle and deleting two points in two different fibers over the base manifold The result is a new fibered manifold where all the fibers except two are connected. Properties Any surjective submersion is open: for each open the set is open in Each fiber is a closed embedded submanifold of of dimension A fibered manifold admits local sections: For each there is an open neighborhood of in and a smooth mapping with and A surjection is a fibered manifold if and only if there exists a local section of (with ) passing through each 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