Properties
Let (resp. ) be an n-dimensional (resp. p-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 with and with The above fiber chart condition may be equivalently expressed by
where is the first projection. The chart is then obviously unique. In view of the above property, the 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
Any surjective submersion is open: for each open the set is open in .
A fibered manifold admits local sections: For each there is an open neighborhood of in and a smooth mapping with and .
Read more about this topic: Fibered Manifold
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)