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How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature?

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Well, the obvious thing to do is to take geodesic polar coordinates. Have you tried that? You should be able to prove that constant Gaussian curvature forces the metric to take a particular form, and this will in turn imply that geodesic circles have constant geodesic curvature. –  Zhen Lin May 11 '11 at 14:14

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