Abstract
In three dimensional Riemannian space forms, introducing a natural moving frame, we define the associate curve of a curve and study the properties and relations of a curve and its associate curve. We state necessary and sufficient condition that a Frenet curve is a Bertrand curve in three dimensional Riemannian space forms, especially in a Riemannian 3-dimensional sphere and in a 3-dimensional hyperbolic space, resp. At the same time we give an explicit expression of the partner curve of a Bertrand curve.
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H. Liu was supported by NSFC (No. 11371080), Joint Research of NSFC and NRF, and partially supported by the Chern Institute of Mathematics and Northeastern University.
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Liu, H. Curves in Three Dimensional Riemannian Space Forms. Results. Math. 66, 469–480 (2014). https://doi.org/10.1007/s00025-014-0387-1
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DOI: https://doi.org/10.1007/s00025-014-0387-1