If you embed a manifold $M$ in Euclidean space, is the normal bundle always trivial? Or give an example with non-trivial normal bundle.


If $M$ is non-orientable, its normal bundle must also be non-orientable (or you could use it to find an orientation on $M$ itself).

  • $\begingroup$ But I think tubular neighborhood of open Mobius strip in R^3 is open solid torus. $\endgroup$ – Paladin Jun 7 '19 at 6:49
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    $\begingroup$ @Neel: The underlying space is an open solid torus, but it's non-orientable (and therefore nontrivial) as a bundle over the Mobius strip. $\endgroup$ – Micah Jun 7 '19 at 17:50

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