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I would like to know if there is a theorem along those lines: let $V$ be a submanifold in $\mathbb{R}^n$ such that $V$ is the boundary of a submanifold with boundary $W$. Then, the normal bundle of $V$ is trivial.Maybe there are conditions of dimension or orientability for the statment to be true?

The paradigmatic exemple I have in mind is the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^n$ see as the boundary of the closed ball $B^n$ but I would like to know if there is a general pattern behind that.

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    $\begingroup$ The normal bundle of the boundary in a manifold with boundary is trivial. Hence the normal bundle of $V$ in $\mathbb{R}^n$ splits of a trivial bundle. Thus if $V$ has codimension $1$ your claim is clearly true. $\endgroup$
    – Thomas Rot
    Commented Jun 9, 2017 at 9:54

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