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On some curves in 3-dimensional hyperbolic geometry and solvgeometry

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Abstract

We study curve geometry in para-Sasakian 3-manifolds, especially in the hyperbolic 3-space and the space \(\mathrm {Sol}_3\) of solvgeometry. Parametric expression for \(\varphi \)-trajectories in the hyperbolic 3-space is given.

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References

  1. Adati, T., Matsumoto, K.: On conformally recurrent and conformally symmetric P-Sasakian manifolds. TRU Math. 13(1), 25–32 (1977)

    MathSciNet  MATH  Google Scholar 

  2. Adati, T., Miyazawa, T.: Some proporties of P-Sasakian manifolds. TRU Math. 13(1), 33–42 (1977)

    MathSciNet  MATH  Google Scholar 

  3. Bejan, C.L., Druţă-Romaniuc, S.L.: F-geodesics on manifolds. Filomat 29(10), 2367–2379 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bejan, C.L., Kowalski, O.: On a generalization of geodesic and magnetic curves. Note Mat. 37(suppl. 1), 49–57 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Bejan, C.L., Meriç, ŞE., Kılıç, E.: Legendre curves on generalized paracontact metric manifolds. Bull. Malays. Math. Sci. Soc. 42, 185–199 (2019)

    Article  MathSciNet  Google Scholar 

  6. Cǎlin, C., Crasmareanu, M., Munteanu, M.I.: Slant curves in three-dimensional f-Kenmotsu manifolds. J. Math. Anal. Appl. 394, 400–407 (2012)

    Article  MathSciNet  Google Scholar 

  7. Calvaruso, G., Munteanu, M.I., Perrone, A.: Killing magnetic curves in three-dimensional almost paracontact manifolds. J. Math. Anal. Appl. 426, 423–439 (2015)

    Article  MathSciNet  Google Scholar 

  8. Chen, B.Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific, Singapore (2017)

    Book  Google Scholar 

  9. Dobarro, F., Ünal, B.: Curvature of multiply warped products. J. Geom. Phys. 55, 75–106 (2005)

    Article  MathSciNet  Google Scholar 

  10. Ikawa, T., Nemoto, H.: On an invariant homogeneous hypersurface in a hyperbolic space as an SP-Sasakian manifold. Tensor (N.S.) 39, 1–4 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Ikawa, T., Nemoto, H.: On P-Sasakian hypersurfaces in a real space form. Tensor (N.S.) 40(2), 107–112 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Inoguchi, J., Lee, J.-E.: \(\varphi \)-trajectories in Kenmotsu manifolds, J. Geom. 113 (2022), Article number 8

  13. Kaneyuki, S., Williams, F.L.: Almost paracontact and parahodge structures on manifolds. Nagoya Math. J. 99, 173–187 (1985)

    Article  MathSciNet  Google Scholar 

  14. Lee, J.E.: Slant curves and biharmonic Frenet curves in 3-dimensional para-Sasakian manifolds. Balkan J. Geom. Appl. 26(1), 21–33 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Manev, M., Staikova, M.: On almost paracontact Riemannian manifolds of type (n, n). J. Geom. 72, 108–114 (2001)

    Article  MathSciNet  Google Scholar 

  16. Matsumoto, K.: On a structure defined by a tensor field\(f\)of type\((1,1)\)satisfying\(f^{3}-f=0\). Bull. Yamagata Univ. Natur. Sci. 9(1), 33–46 (1976)

  17. Matsumoto, K.: A certain vector field in a compact orientable P-Sasakian manifold. Bull. Yamagata Univ. Natur. Sci. 9(2), 211-215 (1976/77)

  18. Matsumoto, K.: Conformal Killing vector fields in a P-Sasakian manifold. J. Korean Math. Soc. 14(1), 135-142 (1977/78)

  19. Matsumoto, K.: On conformal P-Killing vector fields in almost paracontact Riemannian manifolds. J. Korean Math. Soc. 18(1), 73-80 (1981/82)

  20. Matsumoto, K.: On infinitesimal curvature-preserving variations of a P-Sasakian hypersurface in a locally product Riemannian manifold. Bull. Yamagata Univ. Natur. Sci. 10(3), 265–272 (1982)

    MathSciNet  Google Scholar 

  21. Nemoto, H.: On almost paracontact manifolds. TRU Math. 17(1), 89–102 (1981)

    MathSciNet  MATH  Google Scholar 

  22. Nistor, A.I.: New examples of F-planar curves. Kragujevac J. Math. 43(2), 247–257 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Ogata, T.: On infinitesimal transformations of a P-Sasakian manifold. Bull. Yamagata Univ. Natur. Sci. 9(3), 341–349 (1978)

    MathSciNet  Google Scholar 

  24. Sasaki, S.: On paracontact Riemannian maifolds. TRU Math. 16(2), 75–86 (1981)

    Google Scholar 

  25. Satō, I.: On a structure similar to almost contact structure. Tensor (N.S.) 30(3), 219–224 (1976)

    MathSciNet  MATH  Google Scholar 

  26. Satō, I.: On a structure similar to almost contact structure II. Tensor (N.S.) 31(2), 199–205 (1977)

    MathSciNet  MATH  Google Scholar 

  27. Satō, I.: On a Riemannian manifold admitting a certain vector field. Kōdai Math. Sem. Rep. 29, 250–260 (1978)

    MathSciNet  MATH  Google Scholar 

  28. Satō, I., Matsumoto, K.: On P-Sasakian manifolds satisfying certain conditions. Tensor (N.S.) 33(2), 173–178 (1979)

    MathSciNet  MATH  Google Scholar 

  29. Thurston, W. M.: Three-dimensional Geometry and Topology I, Princeton Math. Series., vol. 35 (S. Levy ed.), 1997.

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Acknowledgements

The author is partially supported by JSPS KAKENHI Grant Number JP19K03461.

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, 305-8571, Japan

    Jun-ichi Inoguchi

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  1. Jun-ichi Inoguchi
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Correspondence to Jun-ichi Inoguchi.

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Dedicated to professor Koji Matsumoto on the occasion of his 80th birthday.

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Inoguchi, Ji. On some curves in 3-dimensional hyperbolic geometry and solvgeometry. J. Geom. 113, 37 (2022). https://doi.org/10.1007/s00022-022-00650-6

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  • DOI: https://doi.org/10.1007/s00022-022-00650-6

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