Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SHARP INEQUALITIES INVOLVING THE CHEN-RICCI INEQUALITY FOR SLANT RIEMANNIAN SUBMERSIONS

  • Mehmet Akif Akyol (Department of Mathematics Faculty of Arts and Sciences Bingol University) ;
  • Nergiz (Onen) Poyraz (Department of Mathematics Faculty of Arts and Sciences Cukurova University)
  • Received : 2022.07.25
  • Accepted : 2023.06.16
  • Published : 2023.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Main objective of the present paper is to establish Chen inequalities for slant Riemannian submersions in contact geometry. In this manner, we give some examples for slant Riemannian submersions and also investigate some curvature relations between the total space, the base space and fibers. Moreover, we establish Chen-Ricci inequalities on the vertical and the horizontal distributions for slant Riemannian submersions from Sasakian space forms.

Keywords

References

  1. M. A. Akyol, Conformal semi-slant submersions, Int. J. Geom. Methods Mod. Phys. 14 (2017), no. 7, 1750114, 25 pp. https://doi.org/10.1142/S0219887817501146
  2. M. A. Akyol and R. Prasad, Semi-slant ξ⊥-, hemi-slant ξ⊥-Riemannian submersions and quasi hemi-slant submanifolds, In: Chen, B. Y., Shahid, M. H., Al-Solamy, F. (eds) Contact Geometry of Slant Submanifolds. Springer, Singapore.
  3. M. A. Akyol and R. Sari, On semi-slant ξ⊥-Riemannian submersions, Mediterr. J. Math. 14 (2017), no. 6, Paper No. 234, 20 pp. https://doi.org/10.1007/s00009-017-1035-2
  4. P. Alegre, B.-Y. Chen, and M. I. Munteanu, Riemannian submersions, δ-invariants, and optimal inequality, Ann. Global Anal. Geom. 42 (2012), no. 3, 317-331. https://doi.org/10.1007/s10455-012-9314-4
  5. M. E. Aydin, A. Mihai, and I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat 29 (2015), no. 3, 465-476. https://doi.org/10.2298/FIL1503465A
  6. H. Aytimur and C. Ozgur, Sharp inequalities for anti-invariant Riemannian submersions from Sasakian space forms, J. Geom. Phys. 166 (2021), Paper No. 104251, 12 pp. https://doi.org/10.1016/j.geomphys.2021.104251
  7. D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer, Berlin, 1976.
  8. J.-P. Bourguignon, A mathematician's visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino 1989 (1990), Special Issue, 143-163.
  9. J.-P. Bourguignon and H. B. Lawson Jr., Stability and isolation phenomena for Yang-Mills fields, Comm. Math. Phys. 79 (1981), no. 2, 189-230. http://projecteuclid.org/euclid.cmp/1103908963 103908963
  10. 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
  11. B.-Y. Chen, Slant immersions, Bull. Austral. Math. Soc. 41 (1990), no. 1, 135-147. https://doi.org/10.1017/S0004972700017925
  12. B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel) 60 (1993), no. 6, 568-578. https://doi.org/10.1007/BF01236084
  13. B.-Y. Chen, A general inequality for submanifolds in complex-space-forms and its applications, Arch. Math. (Basel) 67 (1996), no. 6, 519-528. https://doi.org/10.1007/BF01270616
  14. B.-Y. Chen, Riemannian submersions, minimal immersions and cohomology class, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 162-167 (2006). http://projecteuclid.org/euclid.pja/1135791768
  15. B.-Y. Chen, Examples and classification of Riemannian submersions satisfying a basic equality, Bull. Austral. Math. Soc. 72 (2005), no. 3, 391-402. https://doi.org/10.1017/S000497270003522X
  16. B.-Y. Chen, Pseudo-Riemannian Geometry, δ-Invariants and Applications, World Sci. Publ., Hackensack, NJ, 2011. https://doi.org/10.1142/9789814329644
  17. B.-Y. Chen, A. Mihai, and I. Mihai, A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature, Results Math. 74 (2019), no. 4, Paper No. 165, 11 pp. https://doi.org/10.1007/s00025-019-1091-y
  18. I. K. Erken and C. Murathan, Slant Riemannian submersions from Sasakian manifolds, Arab J. Math. Sci. 22 (2016), no. 2, 250-264. https://doi.org/10.1016/j.ajmsc.2015.12.002
  19. M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian Submersions and Related Topics, World Sci. Publishing, Inc., River Edge, NJ, 2004.
  20. A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715-737.
  21. M. Gulbahar, S. Eken Meric, and E. Kilic, Sharp inequalities involving the Ricci curvature for Riemannian submersions, Kragujevac J. Math. 41 (2017), no. 2, 279-293. https://doi.org/10.5937/kgjmath1702279g
  22. M. Gulbahar, E. Kilic, S. K. Keles, and M. M. Tripathi, Some basic inequalities for submanifolds of nearly quasi-constant curvature manifolds, Differ. Geom. Dyn. Syst. 16 (2014), 156-167.
  23. R. S. Gupta, B. Y. Chen's inequalities for bi-slant submanifolds in cosymplectic space forms, Sarajevo J. Math. 9(21) (2013), no. 1, 117-128. https://doi.org/10.5644/SJM.09.1.11
  24. S. Ianu,s and M. Vi,sinescu, Kaluza-Klein theory with scalar fields and generalised Hopf manifolds, Classical Quantum Gravity 4 (1987), no. 5, 1317-1325. http://stacks.iop.org/0264-9381/4/1317 https://doi.org/10.1088/0264-9381/4/5/026
  25. S. Ianus and M. Visinescu, Space-time compactification and Riemannian submersions, in The mathematical heritage of C. F. Gauss, 358-371, World Sci. Publ., River Edge, NJ, 1991.
  26. E. Kilic, M. M. Tripathi, and M. Gulbahar, Chen-Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. 116 (2016), no. 1, 37-56. https://doi.org/10.4064/ap3666-12-2015
  27. G. Koprulu and B. Sahin, Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers, Int. J. Geom. Methods Mod. Phys. 18 (2021), no. 11, Paper No. 2150169, 11 pp. https://doi.org/10.1142/S0219887821501693
  28. J. W. Lee, Anti-invariant ξ⊥-Riemannian submersions from almost contact manifolds, Hacet. J. Math. Stat. 42 (2013), no. 3, 231-241.
  29. A. Mihai and C. Ozgur, Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections, Rocky Mountain J. Math. 41 (2011), no. 5, 1653-1673. https://doi.org/10.1216/RMJ-2011-41-5-1653 
  30. I. Mihai and I. Presura, An improved first Chen inequality for Legendrian submanifolds in Sasakian space forms, Period. Math. Hungar. 74 (2017), no. 2, 220-226. https://doi.org/10.1007/s10998-016-0161-0
  31. 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
  32. B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469. http://projecteuclid.org/euclid.mmj/1028999604 1028999604
  33. K.-S. Park and R. Prasad, Semi-slant submersions, Bull. Korean Math. Soc. 50 (2013), no. 3, 951-962. https://doi.org/10.4134/BKMS.2013.50.3.951
  34. N. (Onen) Poyraz, Chen inequalities on spacelike hypersurfaces of a GRW spacetime, Differential Geom. Appl. 81 (2022), Paper No. 101863, 11 pp. https://doi.org/10.1016/j.difgeo.2022.101863
  35. B. S,ahin, 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
  36. B. S,ahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102) (2011), no. 1, 93-105.
  37. B. S,ahin, Chen's first inequality for Riemannian maps, Ann. Polon. Math. 117 (2016), no. 3, 249-258. https://doi.org/10.4064/ap3958-7-2016
  38. B. Sahin, Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and Their Applications, Elsevier/Academic Press, London, 2017.
  39. C. Sayar, M. A. Akyol, and R. Prasad, Bi-slant submersions in complex geometry, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 4, 2050055, 17 pp. https://doi.org/10.1142/S0219887820500553
  40. M. M. Tripathi, Certain basic inequalities for submanifolds in (κ, µ)-spaces, in Recent advances in Riemannian and Lorentzian geometries (Baltimore, MD, 2003), 187-202, Contemp. Math., 337, Amer. Math. Soc., Providence, RI, 2003. https://doi.org/10.1090/conm/337/06061
  41. G.-E. Vˆilcu, On Chen invariants and inequalities in quaternionic geometry, J. Inequal. Appl. 2013 (2013), 66, 14 pp. https://doi.org/10.1186/1029-242X-2013-66
  42. B. Watson, Almost Hermitian submersions, J. Differential Geometry 11 (1976), no. 1, 147-165. http://projecteuclid.org/euclid.jdg/1214433303 https://doi.org/10.4310/jdg/1214433303
  43. B. Watson, G, G' -Riemannian submersions and nonlinear gauge field equations of general relativity, in Global analysis-analysis on manifolds, 324-349, Teubner-Texte Math., 57, Teubner, Leipzig, 1983.
  44. L. Zhang and P. Zhang, Notes on Chen's inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection, J. East China Norm. Univ. Natur. Sci. Ed. 2015 (2015), no. 1, 6-15, 26.