I'm learning differential geometry,and need help with the following question:
Does there exist a constant curvature curve on paraboloid surface $ z = x^2 + y^2$ different from plane curve, i.e $z$ does not constant?
2026-03-27 15:18:03.1774624683
Does there exist a constant curvature curve on paraboloid surface?
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Look at a constant-$z>0$ circle. It has both constant geodesic curvature on the paraboloid, and constant curvature as space curve.