Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON SLANT RIEMANNIAN SUBMERSIONS FOR COSYMPLECTIC MANIFOLDS

  • Erken, Irem Kupeli (Department of Mathematics Faculty of Art and Science Uludag University) ;
  • Murathan, Cengizhan (Department of Mathematics Faculty of Art and Science Uludag University)
  • Received : 2013.11.15
  • Published : 2014.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifold. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submersions in the cases where the characteristic vector field is vertical or horizontal.

Keywords

References

  1. P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press, Oxford, 2003.
  2. D. E. Blair, Contact Manifolds in Riemannian Geometry, Lectures Notes in Mathematics 509, Springer-Verlag, Berlin, 1976.
  3. J. P. Bourguignon and H. B. Lawson, Stability and isolation phenomena for Yang-mills fields, Comm. Math. Phys. 79 (1981), no. 2, 189-230. https://doi.org/10.1007/BF01942061
  4. J. P. Bourguignon and H. B. Lawson, Mathematician's visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino (1989), Special Issue, 143-163.
  5. J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, and M. Fernandez, Slant submanifolds in Sasakian Manifolds, Glasg. Math. J. 42 (2000), no. 1, 125-138. https://doi.org/10.1017/S0017089500010156
  6. B. Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Leuven, 1990.
  7. D. Chinea, Almost contact metric submersions, Rend. Circ. Mat. Palermo (2) 34 (1985), no. 1, 89-104. https://doi.org/10.1007/BF02844887
  8. R. H. Escobales Jr., Riemannian submersions with totally geodesic fibers, J. Differential Geom. 10 (1975), 253-276.
  9. A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715-737.
  10. S. Ianus, A. M. Ionescu, R. Mazzocco, and G. E. Vilcu, Riemannian submersions from almost contact metric manifolds, arXiv: 1102.1570v1 [math. DG]. https://doi.org/10.1007/s12188-011-0049-0
  11. S. Ianus, R. Mazzocco, and G. E. Vilcu, Riemannian submersions from quaternionic manifolds, Acta Appl. Math. 104 (2008), no. 1, 83-89. https://doi.org/10.1007/s10440-008-9241-3
  12. S. Ianus and M. Visinescu, Kaluza-Klein theory with scalar fields and generalized Hopf manifolds, Class. Quantum Gravity 4 (1987), 1317-1325. https://doi.org/10.1088/0264-9381/4/5/026
  13. S. Ianus and M. Visinescu, Space-time compactification and Riemannian submersions, In: Rassias, G.(ed.) The Mathematical Heritage of C. F. Gauss, (1991), 358-371, World Scientific, River Edge.
  14. B. H. Kim, Fibred Riemannian spaces with quasi Sasakian structure, Hiroshima Math. J. 20 (1990), no. 3, 477-513.
  15. I. Kupeli Erken and C. Murathan, Slant Riemannian submersions from Sasakian manifolds, arXiv: 1309.2487v1 [math. DG].
  16. G. D. Ludden, Submanifolds of cosymplectic manifolds, J. Differential Geom. 4 (1970), 237-244.
  17. C. Murathan and I. Kupeli Erken, Anti-invariant Riemannian submersions from cosymplectic manifolds, arXiv:1302.5108v1 [math. DG].
  18. M. T. Mustafa, Applications of harmonic morphisms to gravity, J. Math. Phys. 41 (2000), no. 10, 6918-6929. https://doi.org/10.1063/1.1290381
  19. Z. Olszak, On almost cosymplectic manifolds, Kodai Math. J. 4 (1981), no. 2, 239-250. https://doi.org/10.2996/kmj/1138036371
  20. B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469. https://doi.org/10.1307/mmj/1028999604
  21. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York-London 1983.
  22. K. S. Park, H-slant submersions, Bull. Korean Math. Soc. 49 (2012), no. 2, 329-338. https://doi.org/10.4134/BKMS.2012.49.2.329
  23. K. S. Park, H-semi-invariant submersions, Taiwanese J. Math. 16 (2012), no. 5, 1865-1878.
  24. B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Cent. Eur. J. Math. 8 (2010), no. 3, 437-447. https://doi.org/10.2478/s11533-010-0023-6
  25. B. Sahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie (N.S) 54(102) (2011), no. 1, 93-105.
  26. B. Sahin, Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math. 17 (2013), no. 2, 629-659.
  27. B. Sahin, Semi-invariant submersions from almost Hermitian manifolds, Canad. Math. Bull. 56 (2013), no. 1, 173-183. https://doi.org/10.4153/CMB-2011-144-8
  28. H. M. Tastan, On Lagrangian submersion, arXiv: 1311.1676v1 [math. DG].
  29. B. Watson, Almost Hermitian submersions, J. Differential Geom. 11 (1976), no. 1, 147-165.
  30. B. Watson, G, G'-Riemannian submersions and nonlinear gauge field equations of general relativity, In: Rassias, T. (ed.) Global Analysis-Analysis on manifolds, dedicated M. Morse, 324-349, Teubner-Texte Math., 57, Teubner, Leipzig, 1983.
  31. D. W. Yoon, Inequality for Ricci curvature of slant submanifolds in cosymplectic space forms, Turkish J. Math. 30 (2006), no. 1, 43-56.

Cited by

  1. Conformal semi-invariant submersions vol.19, pp.02, 2017, https://doi.org/10.1142/S0219199716500115
  2. Conformal semi-slant submersions vol.14, pp.07, 2017, https://doi.org/10.1142/S0219887817501146
  3. Semi-invariant submersions whose total manifolds are locally product Riemannian 2017, https://doi.org/10.2989/16073606.2017.1335657
  4. On anti-invariant Riemannian submersions whose total manifolds are locally product Riemannian vol.108, pp.2, 2017, https://doi.org/10.1007/s00022-016-0347-x