While doing some study on geodesics on Riemannian manifolds, I learned that any geodesics on $n$-spheres are part of great circles. I then started wondering if that is true for any spherically symmetric manifold. My thought is, for a manifold $M=N \times S^2$, since this manifold is spherically symmetric I can always choose my coordinate system such that the polar angle is constant for the initial and ending points connected by the geodesic. But how can I prove that along the geodesic the polar angle doesn't change? I'm I wrong in my hypotheses?

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    $\begingroup$ Check out math.stackexchange.com/questions/258017/… $\endgroup$ – Neal Oct 30 '13 at 19:30
  • $\begingroup$ @Neal Well, that settles it. Thank you for your comment, that post didn't pop up when I browsed for an answer. Thank you once again $\endgroup$ – PML Oct 30 '13 at 21:34

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