Does there exist a smooth embedding $\varphi\colon S^k\to S^n$ such that $\varphi(S^k)$ has non-trivial normal bundle? I looked at some of the old papers by Kervaire, Haefliger, Massey, Levine but I couldn't find an answer. There are various related results, for example it is shown by Kervaire et al that in many dimensions the normal bundle is trivial. Furthermore Kervaire showed that there exists an immersion, such that the pullback of the normal bundle is non-trivial.

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