TITLE: The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point
QUESTION [5 upvotes]: Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$  be the space of  all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$.
Then $K$ is  a  convex  closed subset of  all  $C^1$  1-forms on $\mathbb{T}^2$.

Is $K$ a compact  subset of the space of 1-forms with respect to $C^1$ topology? If  the  answer is affirmative. according to the Krein Millman theorem, what is a presise description of its  extreme points of  $K$?


Does the topological structure of $K$ depends on chosing the vector field $X$ tangent to our initial Kronecker foliation of torus? Does the topological structure of $K$ depend on the slope of our Kronecker foliation?

Motivation:
A motivation for this  question is the following:
In this post and  some  other related linked posts we try to  find  a Riemannian metric compatible to orbits of  a  non vanishing vector  fields. Choosing various metrics enable us to have different  curvatuare functions. Possessing  an appropriate curvature function is very essential for appllying the Gauss Bonnet theorem to the problem of limit cycles of  vctor fields.(For counting them as closed geodesics). So this situation leads us to think about the diversity of closed differential 1-forms $\alpha$  with $\alpha(X)=1$. Under these conditions, in particular the propery of  closed convexity of this set $K$.  one is tempt  to be  curious  about the presice description of   possible extrem points of $K$.
Remark: For generalization of  this question to $n$ dimensional space we should consider the spaceof  all 1-form $\alpha$   with $i_X d\alpha=0,\;\alpha(X)=1$.

REPLY [5 votes]: I don't think it is compact, but perhaps I miss a normalization condition?
Let $X=\partial_x+a\partial_y$, with $a$ irrational (doesn't actually matter for the following). Let $\alpha\in C$ (e.g. $\alpha=dx$) and let $\omega_\lambda=\lambda(a dx-dy)$, for $\lambda\in \mathbb R$. As $X$ lives in the kernel of $\omega_\lambda$, and $d\omega_\lambda=0$, then $\alpha+\omega_\lambda\in C$.