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A circle bundle is a <fiber bundle> with fiber $S^1$. Oriented circle bundles over a paracompact base are classified by their <Euler class> in $H^2(B;\mathbb Z)$.


Circle bundle

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Every proper surjective <submersion> is a locally trivial smooth <fiber bundle>. In particular, the fibers in a proper smooth family are diffeomorphic.


Ehresmann fibration theorem

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A fiber bundle is a map $\pi:E\to B$ locally isomorphic over each open set $U\subseteq B$ to the projection $U\times F\to U$ for a fixed fiber $F$.


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 Incoming links: Fiber bundle