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 figurefigure> 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 ):
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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), 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 fugurefigure):
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), 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 fugure):
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), 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):
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 thesesthese 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 closedsimple closed curve which can be counted as a 2-polygon(with two vertex), whose sum angles is $2\pi$, look at the following figure:**
https://commons.m.wikimedia.org/wiki/File(Note that the horizontal axis is assumed to be a geodesic as well as other curves in the fugure):Limit_cycle_Poincare_map.svg
In the above figure all curves are assumed to be geodesics including 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.
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:**
https://commons.m.wikimedia.org/wiki/File:Limit_cycle_Poincare_map.svg
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.
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), 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 fugure):
In the above figure all curves are assumed to be geodesics including the (horizontal) $x$-axis.
