>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]( https://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-or-positively-curved-space) 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 $\kappa_g|X|=X.y$ 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](https://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-or-positively-curved-space).
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](https://mathoverflow.net/questions/273970/a-cubic-system-with-two-nested-limit-cycles-with-opposite-orientations/274981#274981)  of  a  polynomial  vector  field  whose  flow  is  not  geodesible.  Does this  system generate a  WG flow?