I expect that the answer my question is some standard theorem that I must know but I do not know:)
Assume that we have a smooth embedding $f:M\to\mathbb{R}^m=\{(x^1,\ldots,x^m)\}$ of a smooth manifold $M$ into $\mathbb{R}^m,\quad n=\mathrm{dim}\, M<m$.
Which are the sufficient conditions that guarantee that the image $f(M)$ can be presented as a system of equations $$g_i(x)=0,\quad i=1,\ldots,m-n,\quad \mathrm{rang}\frac{\partial g_i}{\partial x^k}=m-n$$ with functions $g_i$ smooth in a neighbourhood of $f(M)$?
