Communications of the Korean Mathematical Society (대한수학회논문집)

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ON GENERALIZED QUASI-CONFORMAL N(k, μ)-MANIFOLDS

  • Received : 2015.03.25
  • Published : 2016.01.31
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Abstract

The object of the present paper is to introduce a new curvature tensor, named generalized quasi-conformal curvature tensor which bridges conformal curvature tensor, concircular curvature tensor, projective curvature tensor and conharmonic curvature tensor. Flatness and symmetric properties of generalized quasi-conformal curvature tensor are studied in the frame of (k, ${\mu}$)-contact metric manifolds.

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References

  1. Ch. Baikoussis and Th. Koufogiorgos, On a type of contact manifolds, J. Geom. 46 (1993), no. 1-2, 1-9. https://doi.org/10.1007/BF01230994
  2. D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin, 1976.
  3. D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J. (2) 29 (1977), no. 3, 319-324. https://doi.org/10.2748/tmj/1178240602
  4. D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 19 (1995), no. 1-3, 189-214.
  5. E. Boeckx, A full classification of contact metric (k, ${\mu}$)-spaces, Illinois J. Math. 44 (2000), no. 1, 212-219.
  6. E. Cartan, Sur une Classe Remarquable d'espaces de Riemannian, Bull. Soc. Math. France 54 (1926), 214-264.
  7. R. Deszcz and M. G logowska, Some examples of nonsemisymmetric Ricci-semi-symmetric hypersurfaces, Colloq. Math. 94 (2002), no. 1, 87-101. https://doi.org/10.4064/cm94-1-7
  8. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, 1949.
  9. Y. Ishii, On conharmonic transformations, Tensor (N.S.) 7 (1957), 73-80.
  10. O. Kowalski, An explicit classification of three dimensional Riemannian spaces satisfying R(X, Y)${\cdot}$R = 0, Czechoslovak Math. J. 46 (121) (1996), no. 3, 427-474.
  11. K. Nomizu, On the decomposition of generalized curvature tensor fields, Differential geometry (in honor of Kentaro Yano), pp. 335-345. Kinokuniya, Tokyo, 1972.
  12. M. Okumura, Some remarks on space with a certain contact structure, Tohoku Math. J. (2) 14 (1962), 135-145. https://doi.org/10.2748/tmj/1178244168
  13. B. J. Papantoniou, Contact Riemannian manifolds satisfying $R({\xi},X){\cdot}R=0$ and ${\xi}{\in}(k,{\mu})$-nullity distribution, Yokohama Math. J. 40 (1993), no. 2, 149-161.
  14. D. Perrone, Contact Riemannian manifolds satisfying $R({\xi},X){\cdot}R=0$, Yokohama Math. J. 39 (1992), no. 2, 141-149.
  15. G. P. Pokhariyal and R. S. Mishra, Curvatur tensors and their relativistics significance, Yokohama Math. J 18 (1970), 105-108.
  16. A. Sarkar and U. C. De, On the quasi conformal curvature tensor of a (k, ${\mu}$)-contact metric manifold, Math. Rep. (Bucur.) 14(64) (2012), no. 2, 115-129.
  17. S. Sasaki, Almost-contact manifolds Part I Lecture Notes, Mathematical Institue, Tohoku University, 1965.
  18. K. Sekigawa and S. Tanno, Sufficient condition for a Riemannian manifold to be locally symmetric, Pacific J. Math. 34 (1970), 157-162. https://doi.org/10.2140/pjm.1970.34.157
  19. A. A. Shaikh, K. Arslan, C. Murathan, and K. K. Baishya On 3-dimensional generalized (k, ${\mu}$)-contact manifolds, Balkan J. Geom. Appl. 12 (2007), no. 1, 122-134.
  20. A. A. Shaikh and K. K. Baishya, On (k, ${\mu}$)-contact metric manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 51 (2005), no. 2, 405-416.
  21. A. A. Shaikh and K. K. Baishya, On (k, ${\mu}$)-contact metric manifolds, Differ. Geom. Dyn. Syst. 8 (2006), 253-261.
  22. Z. I. Szabo, Structure Theorems on Riemannian spaces satisfying R(X, Y)${\cdot}$R = 0. I, J. Differential Geom. 17 (1982), no. 4, 531-582. https://doi.org/10.4310/jdg/1214437486
  23. Z. I. Szabo, Classification and construction of complete hypersurfaces satisfying R(X, Y)${\cdot}$R = 0., Acta Sci. Math. (Szeged) 47 (1984), no. 3-4, 321-348.
  24. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y)${\cdot}$R = 0. II, Geom. Dedicata 19 (1985), no. 1, 65-108.
  25. S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J. (2) 40 (1988), no. 3, 441-448. https://doi.org/10.2748/tmj/1178227985
  26. K. Yano and S. Bochner, Curvature and Betti Numbers, Ann. of Math. Stud. 32, Princeton University Press, 1953.
  27. K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group, J. Differential Geom. 2 (1968), 161-184. https://doi.org/10.4310/jdg/1214428253
  28. A. Yildiz and U. C. De, A classification of (k, ${\mu}$)-contact metric manifold, Commun. Korean Math. Soc. 27 (2012), no. 2, 327-339. https://doi.org/10.4134/CKMS.2012.27.2.327

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