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Dmitrii Korshunov
Dmitrii Korshunov
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Embedded geodesics are usually very rare. On a flat torus there are only countably many embedded geodesics.

For a generic metric on a 2-dimensional sphere there are only three such geodesics. Since $S^2$ is compact, for a geodesic to be embedded is the same as to be simple (closed without self-intersections). It is known that a triaxial ellipsoid admits exactly three simple geodesics (it follows from Jacobi's integrability of the geodesic flow on an ellipsoid). On the other hand, the number of simple geodesics is an upper-semicontinuous function of metric (this, hopefully, follows from the continuous dependence on parameters for solutions of ODE), so most of the metrics admit not more than three simple geodesics.For a generic metric on a 2-dimensional sphere there are only three such geodesics. Since $S^2$ is compact, for a geodesic to be embedded is the same as to be simple (closed without self-intersections). It is known that a triaxial ellipsoid admits exactly three simple geodesics (it follows from Jacobi's integrability of the geodesic flow on an ellipsoid). On the other hand, the number of simple geodesics is an upper-semicontinuous function of metric (this, hopefully, follows from the continuous dependence on parameters for solutions of ODE), so most of the metrics admit not more than three simple geodesics.

Embedded geodesics are usually very rare. On a flat torus there are only countably many embedded geodesics.

For a generic metric on a 2-dimensional sphere there are only three such geodesics. Since $S^2$ is compact, for a geodesic to be embedded is the same as to be simple (closed without self-intersections). It is known that a triaxial ellipsoid admits exactly three simple geodesics (it follows from Jacobi's integrability of the geodesic flow on an ellipsoid). On the other hand, the number of simple geodesics is an upper-semicontinuous function of metric (this, hopefully, follows from the continuous dependence on parameters for solutions of ODE), so most of the metrics admit not more than three simple geodesics.

Embedded geodesics are usually very rare. On a flat torus there are only countably many embedded geodesics.

For a generic metric on a 2-dimensional sphere there are only three such geodesics. Since $S^2$ is compact, for a geodesic to be embedded is the same as to be simple (closed without self-intersections). It is known that a triaxial ellipsoid admits exactly three simple geodesics (it follows from Jacobi's integrability of the geodesic flow on an ellipsoid). On the other hand, the number of simple geodesics is an upper-semicontinuous function of metric (this, hopefully, follows from the continuous dependence on parameters for solutions of ODE), so most of the metrics admit not more than three simple geodesics.

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Dmitrii Korshunov
Dmitrii Korshunov
  • 2.9k
  • 20
  • 27

Embedded geodesics are usually very rare. On a flat torus there are only countably many embedded geodesics.

For a generic metric on a 2-dimensional sphere there are only three such geodesics. Since $S^2$ is compact, for a geodesic to be embedded is the same as to be simple (closed without self-intersections). It is known that a triaxial ellipsoid admits exactly three simple geodesics (it follows from Jacobi's integrability of the geodesic flow on an ellipsoid). On the other hand, the number of simple geodesics is an upper-semicontinuous function of metric (this, hopefully, follows from the continuous dependence on parameters for solutions of ODE), so most of the metrics admit not more than three simple geodesics.