Probably Trivial manifold theory question

Question

I've been searching for the answer and haven't been able to find it. I am not well informed about manifold theory.

I only would like to know: if a manifold is n-dimensional, is it a requirement that it be contained in n+1-dimensional space?

Say a sphere is a 2-dimensional manifold, but it is contained in $\Bbb R^3$

There would be no knot theory if every one-dimensional manifold was required to be contained in $\mathbb{R}^2$. — José Carlos Santos, Jul 03 '17 at 00:01
Is there a theorem where adding a dimension removes singularities? Like 1/x viewed from $ \Bbb R $. — smokeypeat, Jul 08 '17 at 03:12

MSIS · Accepted Answer · 2017-07-03T00:31:10.947

If by contained, you mean embeddable, then the answer is no. As an example, the Klein bottle, a $2$-manifold (meaning a surface) cannot be embedded in $ \mathbb R^3 $. There is a result, though, called the Whitney embedding theorem whereby you can always embed a $k$-manifold in $\mathbb R^{2k} $, though this is not the sharpest bound, i.e., it may be possible to embed a $k$-manifold in $\mathbb R^{2k-n} $ where $2k>n>0 $; as example of this, the $n$-sphere $ S^{n-1}$ can be embedded in $ \mathbb R^n $ ( EDIT: though the $n$-sphere cannot be embedded in $\mathbb R^{n})$.

Probably Trivial manifold theory question

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