Manifold as zero locus of smooth functions I am used to the local chart-definition of a manifold, so a manifold is for me a priori not necessarily given as a subset of some $\mathbb{R}^n$. I have heard that (in some cases?) a manifold can be given as the zero locus of some "cutting-out" smooth functions $\mathbb{R}^n\to \mathbb{R}$. I am aware of Whitney's embedding theorem, but I don't think its proof explicitly requires or provides such "cutting-out" functions. So my questions are:


*

*Is it true for all manifolds that they can be cut out by smooth functions?

*Where can I find a proof of it?

*Can these "cutting-out" functions for the manifold be given explicitly? Of course they are not unique, but any solution would be good for me.

 A: Locally this is always true. If you have an embedded $k$ dimensional submanifold of $\mathbb{R}^n$ then you can write the manifold locally as a graph over it's tangent plane, i.e.
$$M\cap U = \{(x_{k+1} , \dots, x_n) = (f_{k+1}(x_1, \dots,x_k), \dots, f_{n}(x_1, \dots,x_k)\}$$
where the coordinates are adapted so that the first $k$ coordinates span the tangent space of $M$ in a given point and $(x_1, \dots, x_k)$ are taken from some neighbourhood of $0$.
(This is a consequence of the implicit function theorem and a proof can be found in any reasonable analysis text book or in introductions to Differential Geometry or Differential Topology -- the codimension one case in three dimensional Euclidean space is a good starting point to get a feeling for this. In this case the claim should be intuitively clear, in the neighbourhood of a given point each point of the manifold has a unique orthogonal projection onto the tangent plane, and $M$ is the graph of the inverse of this projection, which, again, usually only exists locally) 
Then $M$ is locally just the zero set of 
$$ \Phi(x) = (x_{k+1} , \dots, x_n) - (f_{k+1}(x_1, \dots,x_k), \dots, f_{n}(x_1, \dots,x_k)) $$
Alternatively you can define the (signed, to make it smooth) distance function $$d_M(p) = \text{dist}(p,M)$$
which, in $p$, is defined as the infimum of the lengths of lines from $p$ with the sign chosen depending on a local orientation of $M$ to $M$. It can be shown that this is a smooth function in the neighbourhood of $M$ with maximal rank for the derivative, and then again the implicit function theorem shows that $M$ is the zero set of a smooth function. If $M$ is oriented and compact (or 'nice' at infinity) this procedure allows a global representation of $M$ as a zero set of a function defined in a neighbourhood of $M$.
No proofs though -- check the literature or wait for more ambitious answers ;-)
