The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this current post, we are highly reducing and ignoring the requirement "Geodesibility".
Let $X$ be a nonvanishing vector field on a Riemannian surface $(M,g)$. For $q\in M$, the geodesic curvature of the orbit of$ X$ passing $q$ is denoted by $\kappa_g(q)$.
Definition: A non vanishing vector field $X$ on a $2$-manifold $M$ satisfies WG property (weak geodesible) if there is a Riemannian metric $g$ on $M$ such that the smooth function $\kappa_g(q).|X(q)|$ belongs to the range of the derivation operator $X(u)=X.u=du(X)$. Namely there exist a smooth function $u$ on $M$ with $X.u=\kappa_g |X|$.
Remark 1: Obviously every geodesible flow is a WG flow.
Example: For the standard metric of $\mathbb{R}^2$ and vectot field $X=cos(y)\partial/\partial_x+ sin(y) \partial/\partial_y$ we have $\kappa_g|X|=sin(y)=X.y$ hence $X$ is a WG flow.
Question 1: Is the Vander Pol vector field $(V)$ bellow a WG vector field on $\mathbb{R}^2\setminus \{0\}$?$$(V)\;\;\;\begin{cases} x'=y-(x^3-x)\\y'=-x\end{cases}$$
Question 2: Is there a negative(or positive) curvature Riemannian metric $g$ on the punctured plane for which $\kappa_g |V|$ lies in the range of derivation associated to $(V)$.
Proposition: If the answer to the second question is affirmative, then we obviously have an alternative proof for the fact that $(V)$ has at most one limit cycle.
Proof: If $\gamma_1, \gamma_2$ are two limit cycles then $\int_{\gamma_i} \kappa_gds=\int_{\gamma_i} \kappa_g|V|dt=0$. Applying the Gauss Bonnet theorem to the annular Region bounded by $\gamma_1, \gamma_2$ gives us a contradiction.
Remark 2: As we mentioned in the head of this post, we are trying to find some resolutions to difficulties appeared in the following efforts for consideration of Limit cycles as closed geodesics. So in this more flexible approach we do not require that all orbits would be geodesics.(A very rigid and non generic situation). On the other hand we are trying to find a relation for the problem of investigation of the range of the differential operator associated to a vector field.
Question 3: This answer gives an example of a polynomial vector field whose flow is not geodesible. Does this system generate a WG flow?
