Hallo,


Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as the zero set of some polynomials? If yes, why ? Are there any references? 



Real analytic submanifolds of $\mathbb{R}^{n}$
I agree with the comments it would be very nice if you make your question more precise, what do you mean by "looks locally as"?


By the way, this question reminds me two beautiful results of S. Akbulut and H. King which they proved in 
"On approximating submanifolds by algebraic sets and a solution to the Nash conjecture."
Invent. Math. 107 (1992), no. 1, 87–98. 


I cite the MathSciNet review:


"Theorem I: Let $M\subset{\bf R}^n$ be a compact smooth submanifold; then $M$ is $\epsilon$-isotopic, for any $\epsilon>0$, to a nonsingular real algebraic set of ${\bf R}^{n+1}$. 


Theorem II: If $M\subset{\bf R}^n$ is a compact smooth submanifold, then $M$ is $\epsilon$-isotopic, for any $\epsilon>0$, to the set of nonsingular points of a real algebraic subset of ${\bf R}^n$."


These are global results, they tell you that compact smooth submanifolds are very close (in the isotopic sense as close as you want) to real algebraic subsets. 


Edit: I am not an expert in real algebraic geometry and I don't know if $M$ is $\epsilon$-isotopic, for any $\epsilon>0$, to a nonsingular real algebraic subset of ${\bf R}^n$ (this is a difficult problem).  

What I know is another result of S. Akbulut and H. King in "Transcendental submanifolds of ${\bf R}^n$." Comment. Math. Helv. 68 (1993), no. 2, 308–318. In this paper they prove that there exist manifolds $M\subset\mathbb{R}^n$ which can not be isotoped to the real parts of nonsingular complex algebraic subvarieties of $\mathbb{C}P^n$.


