I am having trouble proving the following statement, which I think is true (and possibly very basic). Let $M$ be a real differentiable manifold of dimension $(n-1)$ sitting inside $\mathbb{R}^n$. Let $W $ be an algebraic set defined by homogeneous forms in $\mathbb{R}[x\_1, ..., x\_n]$ for which $W \subseteq \mathbb{A}\_{\mathbb{\mathbb{C}}}^n$ has dimension $n-2$ (as an algebraic set in $\mathbb{C}^n$). I want to prove that 
$$
M \not \subseteq W \cap \mathbb{R}^n. 
$$ 


I think one reason why I am having trouble proving this (even though I suspect it might be quite basic) may be that I don't have a good understanding on how the dimension of the manifold and dimension of algebraic sets relate to one another. I would greatly appreciate any comments or references. Thank you. 



How can I prove that $(n-1)$-dimensional manifold is not contained in a $(n-2)$-dimensional affine variety?
You seem to be asking whether it is possible that the topological dimension of the set of real points of an algebraic variety is *greater* than its algebraic dimension (in your case, the former is $n-1$ and the latter is $n-2.$) It seems to be a standard fact that this is impossible (inequality in the opposite dimension is quite possible, as in the standard example of $x^2+y^2 = -1.$) The standard reference seems to be:


*Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise*, [**Algorithms in real algebraic geometry**](http://dx.doi.org/10.1007/3-540-33099-2), Algorithms and Computation in Mathematics 10. Berlin: Springer (ISBN 3-540-33098-4/hbk). x, 662 p. (2006). [ZBL1102.14041](https://zbmath.org/?q=an:1102.14041).


