Czechoslovak Mathematical Journal, Vol. 71, No. 1, pp. 155-172, 2021


On generalized Douglas-Weyl Randers metrics

Tayebeh Tabatabaeifar, Behzad Najafi, Mehdi Rafie-Rad

Received May 29, 2019.   Published online October 29, 2020.

Abstract:  We characterize generalized Douglas-Weyl Randers metrics in terms of their Zermelo navigation data. Then, we study the Randers metrics induced by some important classes of almost contact metrics. Furthermore, we construct a family of generalized Douglas-Weyl Randers metrics which are not $R$-quadratic. We show that the Randers metric induced by a Kenmotsu manifold is a Douglas metric which is not of isotropic $S$-curvature. We show that the Randers metric induced by a Kenmotsu or Sasakian manifold is not Einsteinian. By using $D$-homothetic deformation of a Kenmotsu or Sasakian manifold, we construct a family of generalized Douglas-Weyl Randers metrics and show that the Lie group of projective transformations does not act transitively on the set of generalized Douglas-Weyl Randers metrics.
Keywords:  generalized Douglas-Weyl metric; Randers metric; Kenmotsu manifold; Sasakian manifold
Classification MSC:  53B40, 53C60
DOI:  10.21136/CMJ.2020.0241-19

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References:
[1] S. Bácsó, I. Papp: A note on a generalized Douglas space. Period. Math. Hung. 48 (2004), 181-184. DOI 10.1023/B:MAHU.0000038974.24588.83 | MR 2077695 | Zbl 1104.53015
[2] D. Bao, C. Robles: Ricci and flag curvatures in Finsler geometry. A Sampler of Riemann-Finsler Geometry. Mathematical Sciences Research Institute Publications 50. Cambridge University Press, Cambridge (2004), 197-259. MR 2132660 | Zbl 1076.53093
[3] D. E. Blair: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics 203. Birkhäuser, Basel (2010). DOI 10.1007/978-0-8176-4959-3 | MR 2682326 | Zbl 1246.53001
[4] X. Cheng, Z. Shen: Finsler Geometry: An Approach Via Randers Spaces. Springer, Berlin (2012). DOI 10.1007/978-3-642-24888-7 | MR 3015145 | Zbl 1268.53081
[5] M. H. Emamian, A. Tayebi: Generalized Douglas-Weyl Finsler metrics. Iran. J. Math. Sci. Inform. 10 (2015), 67-75. MR 3497134 | Zbl 1336.53086
[6] G. Hall: On the converse of Weyl's conformal and projective theorems. Publ. Inst. Math., Nouv. Sér. 94 (2013), 55-65. DOI 10.2298/PIM1308055H | MR 3137490 | Zbl 1340.53013
[7] I. Hasegawa, V. S. Sabau, H. Shimada: Randers spaces of constant flag curvature induced by almost contact metric structures. Hokkaido Math. J. 33 (2004), 215-232. DOI 10.14492/hokmj/1285766001 | MR 2034815 | Zbl 1062.53014
[8] B. Li, Z. Shen: On Randers metrics of quadratic Riemann curvature. Int. J. Math. 20 (2009), 369-376. DOI 10.1142/S0129167X09005315 | MR 2500075 | Zbl 1171.53020
[9] T. Milkovszki, Z. Muzsnay: On the projective Finsler metrizability and the integrability of Rapcsák equation. Czech. Math. J. 67 (2017), 469-495. DOI 10.21136/CMJ.2017.0010-16 | MR 3661054 | Zbl 06738532
[10] H. G. Nagaraja, D. L. Kiran Kumar, V. S. Prasad: Ricci solitons on Kenmotsu manifolds under $D$-homothetic deformation. Khayyam J. Math. 4 (2018), 102-109. DOI 10.22034/kjm.2018.57725 | MR 3769595 | Zbl 1412.53048
[11] B. Najafi, B. Bidabad, A. Tayebi: On $R$-quadratic Finsler metrics. Iran. J. Sci. Technol., Trans. A, Sci. 4 (2007), 439-443. MR 2525916 | Zbl 1169.53319
[12] B. Najafi, Z. Shen, A. Tayebi: On a projective class of Finsler metrics. Publ. Math. 70 (2007), 211-219. MR 2288477 | Zbl 1127.53017
[13] B. Najafi, A. Tayebi: Some curvature properties of $(\alpha, \beta)$-metrics. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 60 (2017), 277-291. MR 3701890 | Zbl 1399.53034
[14] A. J. Oubiña: New classes of almost contact metric structure. Publ. Math. 32 (1985), 187-193. MR 0834769 | Zbl 0611.53032
[15] Z. Shen: Volume comparison and its applications in Riemann-Finsler geometry. Adv. Math. 128 (1997), 306-328. DOI 10.1006/aima.1997.1630 | MR 1454401 | Zbl 0919.53021
[16] Y. Shen, Y. Yu: On projectively related Randers metrics. Int. J. Math. 19 (2008), 503-520. DOI 10.1142/S0129167X08004789 | MR 2418194 | Zbl 1152.53015
[17] S. Tanno: The topology of contact Riemannian manifolds. Ill. J. Math. 12 (1968), 700-717. DOI 10.1215/ijm/1256053971 | MR 0234486 | Zbl 0165.24703
[18] A. Tayebi, M. Barzegari: Generalized Berwald spaces with $(\alpha, \beta)$-metrics. Indag. Math., New Ser. 27 (2016), 670-683. DOI 10.1016/j.indag.2016.01.002 | MR 3505987 | Zbl 1343.53077
[19] A. Tayebi, B. Najafi: A class of homogeneous Finsler metrics. J. Geom. Phys. 140 (2019), 265-270. DOI 10.1016/j.geomphys.2019.01.006 | MR 3925072 | Zbl 1417.53024
[20] A. Tayebi, E. Peyghan: On a subclass of the generalized Douglas-Weyl metrics. J. Contemp. Math. Anal., Armen. Acad. Sci. 47 (2012), 70-77. DOI 10.3103/S1068362312020033 | MR 3287918 | Zbl 1302.53081
[21] A. Tayebi, H. Sadeghi: On generalized Douglas-Weyl $(\alpha, \beta)$-metrics. Acta Math. Sin., Engl. Ser. 31 (2015), 1611-1620. DOI 10.1007/s10114-015-3418-2 | MR 3397088 | Zbl 1327.53026
[22] A. Tayebi, H. Sadeghi, E. Peyghan: On generalized Douglas-Weyl spaces. Bull. Malays. Math. Sci. Soc. (2) 36 (2013), 587-594. MR 3071751 | Zbl 1272.53067
[23] Y. Wang: Minimal Reeb vector fields on almost Kenmotsu manifolds. Czech. Math. J. 67 (2017), 73-86. DOI 10.21136/CMJ.2017.0377-15 | MR 3632999 | Zbl 1424.53112
[24] H. Xing: The geometric meaning of Randers metrics with isotropic $S$-curvature. Adv. Math., Beijing 34 (2005), 717-730. MR 2213060

Affiliations:   Tayebeh Tabatabaeifar, Behzad Najafi (corresponding author), Amirkabir University of Technology, Tehran Polytechnic, Rasht St, Tehran, Iran, e-mail: t.tabatabaeifar@aut.ac.ir, behzad.najafi@aut.ac.ir; Mehdi Rafie-Rad, University of Mazandaran, Pasdaran St, Babolsar, Mazandaran, Iran e-mail: rafie-rad@umz.ac.ir


 
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