It seems that any closed parallelizable smooth $n$-manifold can be immersed in the $(n+1)$-dimensional Euclidean space (compared to the general $(2n-1)$-dimensional case). Is there any analogous result for embedding?
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All 3--manifolds are parallelisable but most of them do not embed in $\mathbb{R}^4$. Are you looking for a linear bound with constant less than 2, say? – Marco Golla Oct 01 '15 at 08:47
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Yes.All I am asking is whether we can improve on the dim 2n (from Whitney embedding theorem) in case of parallelizable manifolds. – Kds Oct 01 '15 at 09:08
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Orientable $n>1$-manifolds can always be embedded in $(2n-1)$-dimensional space, but surely you can do better than this under the parallelizability assumption. – mme Oct 01 '15 at 17:49
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To my knowledge it is still an open conjecture, that a parallelisable $n$-manifold embeds in Euclidean space of dimension approximatively $3n/2.$ – András Szűcs Dec 22 '15 at 18:06