I am trying to prove the level sets of the function $$ F(x,y) = 2x+2y+\sin(x)+\sin(y) $$ are connected. I noticed that $\|\nabla F(x,y)\|>0$ for every point of the plane and that $F$ is the sum of a linear function with a $2\pi-$periodic one. Because of the never-vanishing gradient property, all the level sets should be diffeomorphic, and it should hence be enough to prove $F^{-1}(0)$ is connected.
For other similar problems, I usually show that I can express the set as the image through a homomorphism of the real line or a connected set. I can not do this here, however.
I considered using "global implicit function theorems" to glue together the local versions, but I do not seem to make this approach work.
Any suggestion?
It's very easy to characterise $F^{-1}(0)$. I suggest we write $$ F(x, y) = f(x) + f(y),$$ where $$ f(t) := 2t + \sin t.$$
Notice that $f$ is (i) one-to-one and (ii) odd.
This implies that $$ F^{-1}(0) = \{ (x, y) \in \mathbb R^2 : x = -y \},$$ which is clearly connected.
And then, what about the other level sets? A Google search about your idea took me to this answer. However, I think we can parametrise the level sets explicitly.
Since $f$ is a continuous bijective function from $\mathbb R$ to $\mathbb R$, it has an inverse $f^{-1}$ which is also a continuous function from $\mathbb R$ to $\mathbb R$.
The level set $F^{-1}(c)$ is given by $$ \{ (f^{-1}(t), f^{-1}(c - t)) : t \in \mathbb R \}.$$
From this representation, hopefully you can see that $F^{-1}(c)$ is homeomorphic to $\mathbb R$, via the maps:
And of course, $\mathbb R$ is connected.
(In fact, this parameterisation is not just a homeomorphism but also a diffeomorphism, since $f^{-1}$ is smooth by virtue of $f$ having a strictly positive derivative everywhere. But that's not relevant for proving that the level sets are connected.)