Let X be a vector field on $\mathbb {T}^2$, we say that $\varphi: \mathbb {R} \to \mathbb {T}^2$ is a periodic orbit of $X $, if $\varphi $ is a periodic function and $\varphi'(t) = X (\varphi(t)), \forall t \in \mathbb {R} $.
Let $H: T\mathbb{T}^2 \to \mathbb{T}^2 \times\mathbb{R}^2$ be a parallelization of $\mathbb{T}^2$, i.e; $H$ is a smooth function such that $H(p,v) = (p, A(p) v)$, where $A(p)$ is a isomophism between $T_p\mathbb{T}^2$ and $\mathbb{R}^2$ definided in the following way:
Consider $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$ (flat torus), then for each point $[(x,y)]\in\mathbb{T}^2$ there are two loops $[(x+t,y)]$ and $[(x,y+t)]$, we define $A([(x,y)])$ as the unique linear transformation between $T_{[(x,y)]} \mathbb {T}^2$ and $\mathbb {R}^2$ satisfying
$$A([(x,y)]) \left.\frac{d}{dt} [(x+t,y)]\right|_{t=0} = (1,0) \quad\text{and,}\quad A([(x,y)]) \left.\frac{d}{dt} [(x,y+t)]\right|_{t=0} = (0,1). $$
I'm having trouble to solve the following exercise (This exercise can be found on page 6 of the book "The dynamics of vector fields in dimension 3-Etienne Ghys").
Exercise: Suppose $X$ is a non-singular vector field on $\mathbb{T}^2$, and consider $X^∗$ as the map $$ X^*: \pi_1\left(\mathbb{T}^2, x_0\right) \to \pi_1 \left(\mathbb{R}^2 \setminus \{0\}, A(x_0)X(x_0) \right)$$ $$\left[\alpha(t)\right] \mapsto \left[A(\alpha(t)) X(\alpha(t))\right] $$ If $X^*$ is a non-trivial homomorphism then $X$ has a periodic orbit.
I was trying to show that if $X$ doesn't admit a periodic orbit, then for every loop $\varphi$ we have $A(\varphi(t)) X(\varphi(t))$ homotopic to a constant curve, but I wasn't able to conclude such thing.
Can anyone help me or can give me some hint?
Update
I noticed that if there exists $\alpha_1,\alpha_2:\mathbb{S}^{1}\to \mathbb{T}^2$, such that $[\alpha_1]$ and $[\alpha_2]$ are generators of $\pi_1(\mathbb{T}^2,x_0)$ and $\{ \alpha_i'(t),X(\alpha_i(t))\}$ is a basis of $T_{\alpha_i(t)} \mathbb{T}^2$, then $A(\alpha_i(t)) X(\alpha_i(t))$ is homotopic to $A(\alpha_i(t)) \alpha'_i (t) $ (the homotopy
$$\tilde{H}(s,t) = A(\alpha_i(s))\left( (1-t) X(\alpha_i(s)) + t \alpha_i'(s) \right) $$
do the job).
And is relatively easy to demonstrate that $A(\alpha_i(t)) \alpha'_i (t)$ is homotopic to the constant map. (I realized that this homotopy doesn't work because it doesn't fix the ends of the interval)
Does anyone know if $X$ is a non-singular vector field on $\mathbb{T}^2$ without periodic orbits then there are $\alpha_1$ and $\alpha_2$ as described above?
I think it's important to inform that in the chapter that this exercise is contained there is the following theorem
Theorem: Every non-singular $\mathcal{C}^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on $\mathbb{T}^2$ with irrational slope
I think that the key to solving the exercise is in the above theorem but I wasn't able to figure out how to do this.
