Cfrgtkky

I found several answers on the following question :

Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary?

All answer says that certainly not. But I cannot understand use of the fact that Boundary of manifold is empty . These answers emphasis on the fact that manifold is compact.

Is it still true that for a compact manifold with boundary there does not exist a no where vanishing exact $1$-form?

The other two examples provide examples showing that you need to assume your manifold is boundaryless. I want to show where the "usual" proof in the boundaryless case breaks down. So, here is the usual proof.

Exactness means $omega = df$ for some smooth $f:Mrightarrow mathbb{R}$. Because $M$ is compact, there is a $pin M$ for which $f(p)$ is an absolute maximum.

We claim that $d_p f = 0$, so that $omega = df$ is not non-vanishing. To show this, we pick $vin T_p M$ and want to show that $(d_p f) v = 0$. To that end, let $gamma:(-epsilon, epsilon)rightarrow M$ be a smooth curve with $gamma(0) = p$ and $gamma'(0) = v$.

We want to show that $(d_p f) v = 0$, or, said another way, that $frac{d}{dt}|_{t=0} f(gamma(t)) = 0$. By definition of derivative, we need to show that $$lim_{hrightarrow 0} frac{f(gamma(h)) - f(gamma(0))}{h} = 0.$$

First, the numerator is $f(gamma(h) - f(p) leq 0$ since $f(p)$ is a maximum of $f$. It follows that $lim_{hrightarrow 0^+} frac{f(gamma(h))- f(p)}{h} leq 0$ and that $lim_{hrightarrow 0^-} frac{ f(gamma(h)) - f(p)}{h} geq 0$.

By assumption, $lim_{hrightarrow 0} frac{f(gamma(h)) - f(p)}{h}$ exists, so the left and right hand limits must match. Since one is non-negative and the other is non-positive, the conclusion is that $lim_{hrightarrow 0} frac{f(gamma(h)) - f(p)}{h} = d_p f v = 0$. $square$

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So, where does this break down if $pin partial M$? Well, tangent vectors on the boundary are defined differently: you only need $gamma$ to have a domain which is $(-epsilon, 0]$ or $[0,epsilon)$.

If we're in the case that the domain of $gamma$ is $[0,infty)$, then the one sided limit $lim_{hrightarrow 0^-} frac{f(gamma(h)) - f(p)}{h}$ no longer makes sense: $gamma(h)$ doesn't make sense for negative $h$, so $f(gamma(h))$ is meaningless.

Likewise, if the domain of $gamma$ is $(-epsilon, 0]$, then $lim_{hrightarrow 0^+} frac{f(gamma(h)) - f(p)}{h}$ no longer makes sense.

Thus, in either case, we lose one of the two inequalities. Without both, the proof no longer works to force $(d_p f) v = 0$. (And the other two answers show there is no way to "fix" the proof to handle the case where $p$ is a boundary point.)

On the compact manifold with boundary $[0,1]$, the differential $1$-form $dx$ is nowhere vanishing and exact. It is the exterior derivative of $x$.

Consider the cylinder, $S^1 times [0, 1]$, parameterized by $theta, t$.

The everywhere nonzero form $dt$ is exact (it's the differential of the function $t$ that gives the "height" coordinate).

So the answer is "no".