Suppose $M$ is a $2n$-dimensional oriented manifold, and $S^{2k-1}\hookrightarrow M$.
Q: Is the normal bundle of $S^{2k-1}$ tirvial?
Is Normal bundle on the odd-dimensional sphere trivial?
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differential-geometry
smooth-manifolds
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0Hint: think of the simplest nonorientable manfiold. – 2017-02-10
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0@JohnMa Sorry, I forgot to say the manifold $M$ is orientable. – 2017-02-10
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1The normal bundle of the diagonal of $M \times M$ is isomorphic to $TM$. – 2017-02-10
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0@MikeMiller So, if $M=S^{2k-1}\times S^{2k-1}$, and $S^{2k-1}\hookrightarrow M$ as the diagonal, then $N(S^{2k-1})\simeq TS^{2k-1}$, which is not trivial in general. Thanks. – 2017-02-11