7 of 11 added 94 characters in body

(Some possible obstructions to ) Limit cycles as closed geodesics(3)

First we explain our Motivation:


First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ and the curvature in the annular region made by theses two closed geodesics has constant sign(either positive or negative).

But the same is true if we keep the closed geodesic $\gamma_1$ and replace the smooth closed geodesic $\gamma_2$ by a non smooth one denoted again by $\gamma_2$. In fact $\gamma_2$ is the union of two pieces of geodesics whose union make a closed curve which can be counted as a 2-polygon(with two vertex), whose sum angles is $2\pi$, look at the following figure:**



In the above figure the (horizontal) $x$-axis is assumed to be a geodesic and we assume that all geodesics in the figure which transversally intersect the x axis, they intersect with the same angle.

Of course the sum angle of the $2$-polygon in the figure is equal to $2\pi$ iff the $x$-axis is an isocline.

On the other hand, in this question, Limit Cycles as closed Geodesics(2), we had intention to consider limit cycles of a quadratic vector field as closed geodesic. Our approach is based on usage of Gauss Bonnete theorem so the sign of Gauss curvature play a crucial role. So we observe that certain "Isocline -type property" of foliation by geodesics could be an obstruction for Looking at limit cycles as closed geodesics in a negatively curved space. We explain this "Isocline- type property" in the main part of our question

[A Picture explaining the question]2

Main part of our question:

Let $M$ be a Riemannian surface with a foliation $\mathcal{F}$ whose leaves are geodesics. We say that $\mathcal{F}$ has the local isocline property if for every $x\in M$ there exist a geodesic $\alpha$ passing $x$ which is locally transverse to the foliation and intersects the leaves of the foliation with a fix angel. We call such a geodesic $\alpha$ an isocline.

Let $X=P\partial_x+Q\partial_y$ be a quadratic polynomial vector field on $\mathbb{R}^2$.

Define the algebraic curve $C=\left\{(x,y)\mid yP-xQ=0\right\}$. Let $W$ be a vector field proportional to $x\partial_x+y\partial_y$.

Associated to the above quadratic vector field, we consider the Riemannian metric on $M=\mathbb{R}^2\setminus C$ whose orthonormal frame is the following: $$\left\{\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}V,\ W\right\}.$$

Then all orbit of the vector field are geodesic of the metric with the above frame. So we have a foliation by geodesics $\mathcal{F}$ on the Riemannian surface $M$.

Question: Is there an example of such situation with the above Local isocline property? As a more general question: Is it true to say that every foliation associated to a quadratic vector field necessarily satisfies the above local property? (A possible positive answer to the latter two questions is an obstruction for consideration of limit cycles as closed geodesics in negatively or positively curved space.) But if the answer is "No", can we choose an appropriate $W$ in the above frame such that this local property fail at all point $x\in M= \mathbb{R}^2\setminus C$?

Note 1: The above definition of "Isocline Local property" is a corrected version of the following post:

An isocline geodesic characterization of $2$ dimensional flat metrics

Note 2: With the same argument as the materials of this post one can prove the following obvious observation:

Obvious Fact:

There is no a conformal Riemannian metric on the punctured plane with non vanishing curvature with the following two properties:

1) The orbits of the solutions of the Van der Pol equation $$\begin{cases}x'=y-(x^3-x)\\ y'=-x \end{cases}$$ are unparametrized geodesics.(Foliation by geodesics)

2)The positive $y$-axis is a geodesic.

Please see this MO question as a related post: Limit cycles as closed geodesics(in negatively or positively curved space)