TITLE: Rigorous justification that overdetermined systems do not have a solution
QUESTION [10 upvotes]: There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions in $N$ variables such that either (a) $n<N$ or (b) $n=N$ and $k<K$. Then this map cannot be onto. Also natural maps in the opposite direction cannot be injective.
I would like to have rigorous theorems making this principle precise.
Let me state few examples of precise statements for which I would like to have a rigorous proof.
Example 1. On an $n$-dimensional manifold $M$, $n>2$, there exists a Riemannian metric which cannot be realized isometrically as a hypersurface in $\mathbb{R}^{n+1}$. (Here we have the obvious map from the space of imbeddings of $M$ to $\mathbb{R}^{n+1}$ to the space of metrics on $M$. The former space is an open subset in the space of $(n+1)$-tuples of functions in $n$ variables; the latter space is the space of $n(n+1)/2$-tuples of functions in $n$ variables. But for $n>2$ one has $n(n+1)/2>n+1$.)
Example 2. Let $n>3$. Consider the map from metrics on $\mathbb{R}^n$ to sections of the tensor bundle $Sym^2(\wedge^2T^*\mathbb{R}^n)$ sending the metric to its Riemann curvature tensor. Then this map cannot be onto. (This example for discussed at this post Equations satisfied by the Riemann curvature tensor)
Example 3. There exist systems of ordinary differential equations of second order on $\mathbb{R}^n$ which cannot be considered as Euler-Lagrange equations for any Lagrangian.
Example 4. Consider the Radon transform between spaces of functions on two real Grassmannians $Gr$ and $Gr'$. If $\dim Gr> \dim Gr'$ then it must have nontrivial kernel.
Finally let me state few rigorous results contradicting the principle stated at the beginning of this post.
1) The Hilbert spaces of $L^2$-functions on any two Riemannian manifolds are isomorphic.
2) The Banach spaces of continuous functions on any two compact manifolds of any positive dimension are isomorphic (it fact, for any uncountable compact metric spaces). (Milyutin's theorem.)
3) The Frechet spaces of infinitely smooth functions on torii of any positive dimension are isomorphic.

REPLY [10 votes]: The principle you mention is not always true ! V. Arnold proved that every continuous function in $N$ real variables is a composition of continuous functions of two variables only. More precisely, there exist $N(2N+1)$ universal functions $\phi_{ij}:[0,1]\rightarrow[0,1]$ such that the map
$$(g_1,\ldots,g_{2N+1})\mapsto\sum_jg_j\left(\sum_i\phi_{ij}(x_i)\right)$$
is onto when acting from $C([0,1])^{2N+1}$ to $C([0,1]^N)$. Here
$$n=1,\quad k=2N+1,\quad K=1.$$
This disproved the conjecture expressed in Hilbert's XIIIth problem.
By the way, is there any detailed proof published in English (I also accept French) ?