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Let $M$ be an arbitrary oriented smooth manifold of dimension $m$. Is it always diffeomorphic to a sumbanifold in ${\mathbb R}^n$ (with some $n$) defined as a set $X$ of common zeroes of $n-m$ smooth functions $f_1,...,f_{n-m}$ (defined on an open set $U\subseteq {\mathbb R}^n$ and having linearly independent differentials in each point $x\in X$: $df_1(x)\wedge...\wedge df_{n-m}(x ) \ne 0$)?

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up vote 2 down vote accepted

If you had such functions then the tangent bundle of the manifold would be stably trivial, which is not always the case. The lowest dimension where there is a countexample is 4. Thus, $\mathbb{CP}^2$ can indeed be embedded in Euclidean space, as can any compact manifold, but its tangent bundle is not stably trivial, in the sense that one cannot add a trivial bundle to the tangent bundle to get another trivial bundle. This follows from characteristic class theory, see e.g. the classic text Milnor, Stasheff, Characteristic classes.

I am not sure what level text you are studying now, so I would add that the gradient vector fields of your functions $f_i$ would give a trivialisation of the normal bundle of the submanifold, and the ambient space $\mathbb{R}^n$ gives a trivialisation of the sum of the tangent bundle and the normal bundle.

See for example this post for a discussion of chern classes of complex projective spaces.

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Is ${\mathbb C}{\mathbb P}^2$ orientable? – Sergei Akbarov Jun 21 '13 at 8:43
Yes. Moreover it is simply connected ($\pi_1(\mathbb{CP}^2)=0$). – Mikhail Katz Jun 21 '13 at 8:44
Interesting... Which book do you recommend? Milnor-Stasheff? – Sergei Akbarov Jun 21 '13 at 8:48
Dear user72694, I think the lowest dimension for a counterexample is $2$, not $4$. Indeed the Stiefel-Whitney class ot the tangent bundle to $\mathbb R \mathbb P^2$ is $1+x+x^2$, so that this tangent bundle cannot be stably trivial. – Georges Elencwajg Jun 21 '13 at 9:12
Those manifolds, which can be represented as zeroes of $f_i$ like I described, do they have special name? – Sergei Akbarov Jun 21 '13 at 9:16

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