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$M$ is a manifold with $m$-dim and there is an embedding $f: M\rightarrow\mathbb{R}^{m+k}$.

My question is

  1. Is following statement true?

$M$ is orientable $\Leftrightarrow $ the normal bundle is trivial

  1. If the statement is not true, is there example $M$ is orientable but normal bundle in Euclidean space is nontrivial? Is there example $M$ is not orientable but normal bundle in Euclidean space is trivial?

PS: I saw an answer there. He gives an example $\mathbb{P}^{2r}\rightarrow \mathbb{R}^{n}$ the normal bundle is nontrivial. But the even dimensional real projective space is non-orientable.

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Every orientable $4$-manifold embeds in $\Bbb R^N$ for some $N$. Take any such 4-manifold $X$ with $w_2(TX) \ne 0$ (e.g. $\Bbb CP^2$). The Whitney sum formula applied to $T\Bbb R^N |_X= TX \oplus NX$ implies $w_2(T\Bbb R^N |_X)=w_2(TX)+w_2(NX)$, hence $w_2(NX)$ is nonzero if $w_2(TX)$ is nonzero.

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    $\begingroup$ On the other hand, the converse is true by basically the same argument: If $NX$ is trivial, then $w_1(TX) = 0$ by the Whitney sum formula, so $X$ is orientable. $\endgroup$ Jan 31 '18 at 22:08

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