Examples of "common" smooth manifolds arrising as vector bundles

Asked Mar 05 '21 at 00:26

Active Mar 05 '21 at 00:26

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Can $S^n$ be realized as a vector bundle of some smooth manifold? What about a Torus? Klein Bottle?

My professor introduced us to fiber bundles today. Am I correct in understanding that we could realize an orientable surface of genus 1 as a a fiber bundle over $S^1$ where each fiber is diffeomorphic to $S^1$?

General insights an examples appreciated.. Thanks MSE!!

asked Mar 05 '21 at 00:26

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This is almost a duplicate of this question. In particular, my answer there shows that the answer is no for all of the proposed manifolds. – Michael Albanese Mar 05 '21 at 00:28
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Well, if you allow rank 0 vector bundles then sure. Otherwise, see Michael Albanese's answer. – Moishe Kohan Mar 05 '21 at 00:32
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Is this about fibre bundles or vector bundles? – Rob Arthan Mar 05 '21 at 00:40
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@MoisheKohan: Not the first time this week I had to be reminded about those. I have edited my answer. – Michael Albanese Mar 05 '21 at 03:04

Examples of "common" smooth manifolds arrising as vector bundles

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