Is there an analytic vector field $X$,on $S^2$ which possess a limit cycle but $X $, satisfy $\nabla_X X =0$ or satisfy $\nabla_X JX= 0$ where $J$ is the standard almost complex structure of $S^2$ and $\nabla$ is the $LC$,connection associated to the standard metric of the sphere ?
$\begingroup$
$\endgroup$
2
asked Feb 14, 2018 at 20:20
-
2$\begingroup$ Because $J$ is covariant constant, i.e., $\nabla J = 0$, one has $\nabla_XJX=J(\nabla_X)$, so the two conditions are the same; they both say that the integral curves of $X$ are geodesics. The only global solution on $S^2$ is $X\equiv0$. $\endgroup$– Robert BryantCommented Feb 15, 2018 at 9:47
-
$\begingroup$ @RobertBryant Thank you for your comment. $\endgroup$– Ali TaghaviCommented Feb 15, 2018 at 21:26
Add a comment
|