I'm reading Loring Tu's "Introduction to Manifolds", and I've came across the following statement (First paragraph of section 11.1, page 115, second edition):
Suppose $ f:N \to M $ Is a smooth map of manifolds, and we want to show that the level set $f^{-1}(c)$ is a manifold for some $c \in M $. In order to apply the regular level set theorem, we need the differential of $f$ to have maximal rank at every point in $f^{-1}(c)$.
I think the statement above is necessary and sufficient if $Dim N \geq Dim M $.
If $Dim N < Dim M$, the maximal rank of $f$ is $Dim N$, and for every point in $f^{-1}(c)$ the differential of $f$ fails to be surjective, so the regular level set theorem is not really an option.
All in all, the statement is necessary, but is not sufficient. Am I wrong?
Best Answer
You are correct that, the way he states it, having maximal rank is necessary but not sufficient. But in applying the regular level set theorem we almost always assume implicitly that $\dim N \ge \dim M$, and I suppose Tu has the same implicit assumption in mind when he wrote that statement.
When $\dim N < \dim M$, the assumption of the regular level set theorem is never satisfied anyway.