Vtyjkyuk

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 $H: TmathbbT^2 to mathbbT^2 timesmathbbR^2$ be a parallelization of $mathbbT^2$, i.e; $H$ is a smooth function such that $H(p,v) = (p, A(p) v)$, where $A(p)$ is a isomophism between $T_pmathbbT^2$ and $mathbbR^2$ definided in the following way:

Consider $mathbbT^2 = mathbbR^2 / mathbbZ^2$ (flat torus), then for each point $[(x,y)]inmathbbT^2$ 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

$$A([(x,y)]) left.fracddt [(x+t,y)]right|_t=0 = (1,0) quadtextand,quad A([(x,y)]) left.fracddt [(x,y+t)]right|_t=0 = (0,1). $$

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 $mathbbT^2$, and consider $X^∗$
as the map $$ X^*: pi_1left(mathbbT^2, x_0right) to pi_1 left(mathbbR^2 setminus 0, A(x_0)X(x_0) right)$$
$$left[alpha(t)right] mapsto left[A(alpha(t)) X(alpha(t))right] $$
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 $alpha_1,alpha_2:mathbbS^1to mathbbT^2$, such that $[alpha_1]$ and $[alpha_2]$ are generators of $pi_1(mathbbT^2,x_0)$ and $ alpha_i'(t),X(alpha_i(t))$ is a basis of $T_alpha_i(t) mathbbT^2$, then $A(alpha_i(t)) X(alpha_i(t))$ is homotopic to $A(alpha_i(t)) alpha'_i (t) $ (the homotopy
$$tildeH(s,t) = A(alpha_i(s))left( (1-t) X(alpha_i(s)) + t alpha_i'(s) right) $$
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 $mathbbT^2$ 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 $mathcalC^2$ vector field on a compact
surface that has no periodic orbits is topologically equivalent to a linear
flow on $mathbbT^2$ 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.