Suppose that $M$ is the interior of a compact manifold with boundary $\partial M$. I'm often faced with so called boundary defining functions - or just 'defining functions'. That are by definition functions $f \in C^{\infty}(\overline{M})$ with the following properties:
(i) $f(x) > 0 \ \forall x \in M$
(ii)$f(x) = 0 \Leftrightarrow x \in \partial M$
(iii) $df \neq 0 \ on \ \partial M$
I need to know what the third property does exactly mean. Often it also is just referred as "has no zeroes on the boundary", "... has a non-vanishing differential there" or "has a non-degenerate differential there".
For me, the differential of a map $f: M \rightarrow N$ at a point p is a linear map $df_p : T_pM \rightarrow T_{f(p)}N$.
Does property (iii) mean: For $v \in T_pM$ with $df_p(v)=0 \Rightarrow v = 0$ (That would be the interpreation: For every point on the boundary, the differential has no other zero than the zero vector.)
Or does it mean: The differential at the point p isn't the zero map, in other words: For every point p on the boundary $\exists v \in T_pM$ with $df_p(v) \neq 0$
Thank you for your help!
Robin