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For unit sphere bundle over sphere there exists a real vector bundle equipped with an inner product structure?

I did't get any results relative to this extension till now. Is there any result regarding this or there is not have such extension?

Please Clarify it..

Thanks

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    $\begingroup$ Could you clarify your question ? Any vector bundle over any manifold admits an inner product... $\endgroup$
    – Nicolast
    Commented Sep 12, 2023 at 19:48
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    $\begingroup$ When you say unit sphere bundle, how is that different from a topological fiber bundle with fibers homeomorphic to spheres? In other words, what is the significance of the word unit? $\endgroup$
    – Ben McKay
    Commented Sep 12, 2023 at 20:05
  • $\begingroup$ We're you thinking of an associated bundle? You can let $O(n+1)$ act on the fibres (assuming the structure group of the fibre bundle is an orthogonal group) and by the defining representation on $\mathbb{R}^{n+1}$, and make the associated vector bundle. If the structure group is homeomorphisms of a sphere, its not clear to me it can be done. $\endgroup$
    – David Roberts ♦
    Commented Sep 13, 2023 at 1:34
  • $\begingroup$ @Nicolast any vector bundle over compact Hausdroff base space admit inner product structure. $\endgroup$
    – Dimpi Paul
    Commented Sep 13, 2023 at 6:23
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    $\begingroup$ Ok I guess I understand your question now: you want to know whether any topological (smooth ?) sphere bundle over manifold (or something else ?) is homeomorphic (diffeomorphic ?) to the unit sphere bundle of a vector bundle with an inner product. Right ? Could you please rephrase your question as it is quite obscure at the moment ? $\endgroup$
    – Nicolast
    Commented Sep 13, 2023 at 7:40

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