The $1$-form $df$ always has zeros

Question

Suppose that $M$ is a compact manifold and $f$ is a smooth function on $M$. Is it true that the $1$-form $df$ always has zeros?

Thank you.

Answer 1

Yes.

$f$, being smooth, is continuous. Therefore, as is well-known, it attains its absolute minimum at some $m_1 \in M$, and its absolute maximum at some $m_2 \in M$; thus

$df(m_1) = 0, \tag 1$

and

$df(m_2) = 0. \tag 2$

Of course, $df$ may have other zeroes on $M$ as well, depending on the topological structure of $M$; this is the subject of Morse Theory.