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It is true that $T^* S^1$ embeds in $\mathbb{R}^2$ as the cotangent of $S^1$ is trivial and $S^1 \times \mathbb{R}$ is diffeomorphic to the punctured plane. Is it true in general that $T^* S^n$ embeds in $\mathbb{R}^{2n}$ ?

Vincent L.
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This is false for $n = 2$. If $T^\ast S^2$ could be embedded in $\Bbb R^4$, then we would also have an embedding of the unit cotangent bundle $ST^\ast S^2 \cong \Bbb R P^3$ in $\Bbb R^4$. But it is known that $\Bbb R P^3$ does not embed in $\Bbb R^4$ (see for example here).

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Henry T. Horton
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