Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LOXODROMES AND TRANSFORMATIONS IN PSEUDO-HERMITIAN GEOMETRY

  • Lee, Ji-Eun (Institute of Basic Science Chonnam National University)
  • Received : 2020.09.14
  • Accepted : 2020.12.22
  • Published : 2021.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we prove that a diffeomorphism f on a normal almost contact 3-manifold M is a CRL-transformation if and only if M is an α-Sasakian manifold. Moreover, we show that a CR-loxodrome in an α-Sasakian 3-manifold is a pseudo-Hermitian magnetic curve with a strength $q={\tilde{r}}{\eta}({\gamma}^{\prime})=(r+{\alpha}-t){\eta}({\gamma}^{\prime})$ for constant 𝜂(𝛄'). A non-geodesic CR-loxodrome is a non-Legendre slant helix. Next, we prove that let M be an α-Sasakian 3-manifold such that (∇YS)X = 0 for vector fields Y to be orthogonal to ξ, then the Ricci tensor 𝜌 satisfies 𝜌 = 2α2g. Moreover, using the CRL-transformation $\tilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\tilde{R}$, the pseudo-Ricci tensor $\tilde{\rho}$ and the torsion tensor field $\tilde{T}^t(\tilde{S}X,Y)$.

Keywords

Acknowledgement

The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1l1A1A01043457).

References

  1. F. R. Al-Solamy, J.-S. Kim, and M. M. Tripathi, On η-Einstein trans-Sasakian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 57 (2011), no. 2, 417-440. https://doi.org/10.2478/v10157-011-0036-x
  2. J. T. Cho, On some classes of almost contact metric manifolds, Tsukuba J. Math. 19 (1995), no. 1, 201-217. https://doi.org/10.21099/tkbjm/1496162808
  3. J. T. Cho, Geometry of CR-manifolds of contact type, in Proceedings of the Eighth International Workshop on Differential Geometry, 137-155, Kyungpook Nat. Univ., Taegu, 2004.
  4. S. Guvenc and C. Ozgur, On pseudo-Hermitian magnetic curves in Sasakian manifolds, preprint, 2020; arXiv:2002.05214v1. https://doi.org/10.22190/FUMI2005291G
  5. J. Inoguchi and J.-E. Lee, Slant curves in 3-dimensional almost contact metric geometry, Int. Electron. J. Geom. 8 (2015), no. 2, 106-146. https://doi.org/10.36890/iejg.592300
  6. J. Inoguchi, M. I. Munteanu, and A. I. Nistor, Magnetic curves in quasi-Sasakian 3-manifolds, Anal. Math. Phys. 9 (2019), no. 1, 43-61. https://doi.org/10.1007/s13324-017-0180-x
  7. S. Koto and M. Nagao, On an invariant tensor under a CL-transformation, Kodai Math. Sem. Rep. 18 (1966), 87-95. http://projecteuclid.org/euclid.kmj/1138845189 https://doi.org/10.2996/kmj/1138845189
  8. J.-E. Lee, Pseudo-Hermitian magnetic curves in normal almost contact metric 3-manifolds, Commun. Korean Math. Soc. 35 (2020), no. 4, 1269-1281. https://doi.org/10.4134/CKMS.c200122
  9. Z. Olszak, Normal almost contact metric manifolds of dimension three, Ann. Polon. Math. 47 (1986), no. 1, 41-50. https://doi.org/10.4064/ap-47-1-41-50
  10. S. Sasaki and Y. Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structure. II, Tohoku Math. J. (2) 13 (1961), 281-294. https://doi.org/10.2748/tmj/1178244304
  11. K. Takamatsu and H. Mizusawa, On infinitesimal CL-transformations of compact normal contact metric spaces, Sci. Rep. Niigata Univ. Ser. A No. 3 (1966), 31-39.
  12. Y. Tashiro and S. Tachibana, On Fubinian and C-Fubinian manifolds, Kodai Math. Sem. Rep. 15 (1963), 176-183. http://projecteuclid.org/euclid.kmj/1138844787 https://doi.org/10.2996/kmj/1138844787
  13. H. Yanamoto, C-loxodrome curves in a 3-dimensional unit sphere equipped with Sasakian structure, Ann. Rep. Iwate Med. Univ. School of Liberal Arts and Science 23 (1988), 51-64.