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Curves in three dimensional Riemannian space forms

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Abstract

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on space forms.

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Shenyang, 110004, People’s Republic of China

    Huili Liu

  2. College of Life Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Yixuan Liu

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  1. Huili Liu
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  2. Yixuan Liu
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Correspondence to Huili Liu.

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Huili Liu supported by NSFC; Joint Research of NSFC and NRF; partially supported by the Chern Institute of Mathematics and Northeastern University.

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Liu, H., Liu, Y. Curves in three dimensional Riemannian space forms. J. Geom. 112, 8 (2021). https://doi.org/10.1007/s00022-021-00574-7

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  • DOI: https://doi.org/10.1007/s00022-021-00574-7

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