To use the regular level set theorem, is it enough to have a maximal rank?
Solution 1:
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.