Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth function $f: \mathbf R^n \to \mathbf R^{n-d}$, submersive at each point of $M$ and such that $M=f^{-1}(0)$.


Of course there are two necessary conditions:
1) $M$ must be a closed subset of $\mathbf R ^n$.
2) The normal bundle of $M$ in $\mathbf R^n$ must be trivial.


At first, I would have guessed that these conditions are sufficient, but I can't prove it.


I have partial answers, however.


1)The first natural thing to do is to take a tubular neighbourhood $U$ of $M$ in $\mathbf R^n$. The indentification $U \simeq M \times \mathbf R^{n-d}$ allows to define a function $f : U \to \mathbf R^{n-d}$ which has the required properties. But it is not clear to me whether $f$ can be extended to the whole $\mathbf R^n$.


2) There is a way to give an answer if we change a bit the problem: the Pontryagin-Thom construction gives a function $f: \mathbf R^n \to \widehat{\mathbf R^{n-d}} \simeq \mathrm S^{n-d}$ by sending all the points outside a tubular neighbourhood at infinity.
This maybe means that this is the good formulation of the problem, but I am still curious about the original one.


3) If $M$ has codimension $1$, then the function $f$ defined on a tubular neighbourhood $U$ of $M$ as in 1) can actually be extended to $\mathbf R ^n$ by a constant function (using the fact that the complement of $M$ has two connected components).



When is a submanifold of $\mathbf R^n$ given by global equations?
An example of a compact $16$-dimensional submanifold of $\mathbb{R}^{30}$ with trivial normal bundle that is not defined by such global equations may be found in [Akbulut-King, Submanifolds and homology of nonsingular real algebraic varieties, [Am J Math **107** 45 (1985)](https://dx.doi.org/10.2307/2374457), Theorem 5.8].


