What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as
$g(x,y)=x^2+y^2-1=0$
for the surface of the unit circle in $\mathbb{R}^2$. What I would like to have happen is to define a manifold by a function $g:\mathbb{R}^m\rightarrow\mathbb{R}^k$ so that the resulting manifold has dimension $m=n-k$ or codimension $k$. The problem that I'm having is that functions can do all sorts of odd things that prevent this. For example, if $g$ defines a space filling curve, we may not restrict the space at all. In this case, we can restrict $g$ to being differentiable and prevent this from happening. Alternatively, it's not always clear if $g:\mathbb{R}^n\rightarrow\mathbb{R}$ peels off a single dimension. For example, if we define
$g(x,y,z)=x^2+y^2+z^2$
we only get a single point, not a 2-dimensional manifold.
As such, I'm looking for assumptions on $g:\mathbb{R}^n\rightarrow\mathbb{R}^k$ that guarantee that we define a manifold of dimension $m-k$ or codimension $k$.
EDIT Notation via Fly By Night
If $g:\mathbb{R}^n\to\mathbb{R}^k$ is regular at some point $x_0\in\mathbb{R}^n$ such that $g(x_0)=0$, then its zero-locus is a (sub)manifold of codimension $k$ in $\mathbb{R}^n$ (basically because of the implicit function theorem).
In the case of $g(x,y,z)=x^2+y^2+z^2$ you have that the zero-locus is given by the single point $(0,0,0)$, where we have $$Dg(0,0,0)=(0,0,0)^T$$ and thus $g$ is not regular there.