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First we explain our Motivation:

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 these two closed geodesics has constant sign(either positive or negative).

But the same is true if we keep the closed geodesic $\gamma_1$, the innermost one, and replace the smooth closed geodesic $\gamma_2$, the outermost one, by a non smooth one denoted again by $\gamma_2$ provided $\angle p+\angle q =2\pi$. (Please see the figure bellow). In fact $\gamma_2$ is the union of two pieces of geodesics whose union make a simple closed curve which can be counted as a 2-polygon(with two vertex $p,q$), whose sum angles is $2\pi$, look at the following figure(Note that the horizontal axis is assumed to be a geodesic as well as other curves in the figure> Furthermore it is obvious that by "angle" we mean Riemannian angle, that is the angles $\angle p$ and $\angle q$ are Riemannian angles not the Euclidean angle drown in the picture ):


GaussBonnete

In the above figure all curves are assumed to be geodesics including the (horizontal) $x$-axis.

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

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)

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  • $\begingroup$ Just a naive comment : have you thought about interpreting limit cycles as the trajectory of a free particle describing a geodesic on a Riemannian manifold moving slowly in some ambient space ? $\endgroup$ – Sylvain JULIEN Jun 10 '18 at 13:14
  • $\begingroup$ @SylvainJULIEN Thank you for your attention. Could you please more explain on your comment. It would be interesting to consider a physical interpretation for this problem. $\endgroup$ – Ali Taghavi Jun 10 '18 at 13:25
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    $\begingroup$ It's just an analogy that may or may not be relevant to you problem. Imagine an ant on Earth going straight ahead from the north pole along the Greenwich meridian till it reaches the south pole, then goes on along the 180° meridian back to north pole at constant speed and there stops. For an observer on Earth the trajectory of the ant is a closed curve. For an observer standing on the Sun, it's not, as the position of the north pole has changed while the Earth revolves our star. $\endgroup$ – Sylvain JULIEN Jun 10 '18 at 14:06
  • $\begingroup$ @SylvainJULIEN your physical approach remind me of two titles which I found many years ago in the web as follows: "Is quantum mechanics a limit cycle theory?" And "Limit cycles in quantum theories". To be honest I do not have a physical background to underestand physical interpretations for limit cycle theiry. Thanks for your attention to my question. $\endgroup$ – Ali Taghavi Jun 11 '18 at 21:53

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