Can someone provide a simple example of the "pre-image theorem" in differential geometry? I only have a background in engineering calculus. A problem I am currently working on relates to something called a "pre-image theorem"
The theorem roughly states:

Let $f: N \to R^{m}$ be a $C^{\infty} $ map on a manifold N of
  dimension n. Then a nonempty regular level set $S = f^{-1}(c)$ is a
  regular submanifold of dimension n-m of N

I am confounded by the language used in this theorem, but I really wish to understand this. Can someone translate this into a simple case where $f$ is some three dimensional function i.e. $f = x^2 + y^2 + z^2$ can show how this theorem applies?
Thanks!
 A: A few examples might make things clear:
(1) Let $n=2$ and $m=1$. Define $f:\mathbb{R}^2 \to \mathbb{R}$ by $f(x,y) = x^2 + y^2$. Then $S = f^{-1}(c) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 =c\}$, which is a circle with radius $\sqrt{c}$. We have $n-m=1$, and a circle is indeed a one-dimensional manifold. 
(2) Let $n=3$ and $m=1$. Define $f:\mathbb{R}^3 \to \mathbb{R}$ by $f(x,y) = x + y + 2z$. Then $S = f^{-1}(c) = \{(x,y,z) \in \mathbb{R}^3 : x + y + 2z = c\}$, which is a plane. We have $n-m=2$, and a plane is indeed a two-dimensional manifold.
(3) Let $n=3$ and $m=2$. Define $f:\mathbb{R}^3 \to \mathbb{R}^2$ by $f(x,y) = (x^2 + y^2, y^2 + z^2)$. Then 
$$
S = f^{-1}(a,b) = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 =a \text{ and } y^2 + z^2 = b\},
$$
which is the curve(s) of intersection of two circular cylinders. We have $n-m=1$, and the set of intersection curves is indeed a one-dimensional manifold.
Since you're not a mathematician, I don't recommend that you spend much time thinking about this theorem. In two or three dimensional space, ignoring all the corner cases and pathologies, all it says is that an equation of the form $f(x,y) = 0$ defines a curve, and an equation of the form $f(x,y,z) = 0$ defines a surface. But those facts are intuitively obvious, anyway, so the theorem doesn't tell us anything new.
