A closed geodesic is a nonconstant periodic geodesic on a Riemannian manifold. It is prime when parametrized with its least positive period, rather than as an iterate of a shorter closed geodesic. Geometric distinctness can be taken to mean distinct images in the manifold. Changing the initial point, reversing orientation, or traversing the same curve repeatedly then gives no new image. Great circles on a round sphere are examples.
Closed Geodesic
See also
Geodesic, Great Circle, Riemannian MetricExplore with Wolfram|Alpha
References
Gallier, J. and Quaintance, J. Differential Geometry and Lie Groups: A Computational Perspective. Oct. 9, 2025. https://www.seas.upenn.edu/~jean/diffgeom-spr-I.pdf.Cite this as:
Weisstein, Eric W. "Closed Geodesic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedGeodesic.html