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Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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Why should I prefer bundles to (surjective) submersions?

The essential point is that a submersion is not necessarily locally trivial whilst this is a crucial assumption for fibre bundles. Necessary and sufficient conditions can be given to ensure the submer …
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