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I know that in Spherical geometry, a rotation is the same as a translation. So a glide reflection is the same as a rotation-reflection or translation-reflection. Also, geodesics in $S^2$, are great circles and if the points are antipodal, then there are infinite number of great circles between them. But I'm not sure how to go about the proof for this question.

user73869
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If a symmetry of the sphere fixes 2 lines, then it must either fix or interchange their 2 points of intersection. That should narrow down the possibilities.

Ted
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  • I'm thinking of an example where there are two great circles, one the equator and one going through the poles. Now I can see their intersection points being fixed and being interchangeable due to the antipodal map. But I'm not sure I understand how this helps with the proof. – user73869 May 06 '13 at 18:50
  • Suppose the 2 intersection points are P and P'. Assume first that they are fixed under some symmetry. Pick a point Q on one of the 2 lines other than P, P'. Where could it go under the symmetry? – Ted May 06 '13 at 21:58