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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
    
@Zhen Lin: Why is it not possible to simply/algebraically express geodesic curvature in terms of Gaussian curvature and $u_0 $? koletennbert's question has not yet been answered. math.stackexchange.com/questions/162474/… – Narasimham Sep 25 '14 at 4:30
    
I expected to arrive at a result maybe somewhat like $ u_0 K +1/u_0.$ – Narasimham Sep 25 '14 at 5:04

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