Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE V-SEMI-SLANT SUBMERSIONS FROM ALMOST HERMITIAN MANIFOLDS

  • Received : 2020.04.30
  • Accepted : 2020.08.03
  • Published : 2021.01.31
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Abstract

In this paper, we deal with the notion of a v-semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. We investigate the integrability of distributions, the geometry of foliations, and a decomposition theorem. Given such a map with totally umbilical fibers, we have a condition for the fibers of the map to be minimal. We also obtain an inequality of a proper v-semi-slant submersion in terms of squared mean curvature, scalar curvature, and a v-semi-slant angle. Moreover, we give some examples of such maps and some open problems.

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References

  1. P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, 29, The Clarendon Press, Oxford University Press, Oxford, 2003. https://doi.org/10.1093/acprof:oso/9780198503620.001.0001
  2. J.-P. Bourguignon, A mathematician's visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino 1989 (1989), Special Issue, 143-163 (1990).
  3. J.-P. Bourguignon and H. B. Lawson, Jr., Stability and isolation phenomena for YangMills fields, Comm. Math. Phys. 79 (1981), no. 2, 189-230. http://projecteuclid.org/euclid.cmp/1103908963 https://doi.org/10.1007/BF01942061
  4. B. Chen, Differential geometry of real submanifolds in a Kahler manifold, Monatsh. Math. 91 (1981), no. 4, 257-274. https://doi.org/10.1007/BF01294767
  5. B. Chen, Geometry of slant submanifolds, Katholieke Universiteit Leuven, Louvain, 1990.
  6. M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian Submersions and Related Topics, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. https://doi.org/10.1142/9789812562333
  7. C. F. Gauss, Disquisitiones generales circa superficies curvas, 1827, http://gdz.sub.unigoettingen.de/no cache/dms/load/img/?IDDOC=139389.
  8. A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715-737.
  9. S. Ianus, A. M. Ionescu, R. Mocanu, and G. E. Vilcu, Riemannian submersions from almost contact metric manifolds, Abh. Math. Semin. Univ. Hambg. 81 (2011), no. 1, 101-114. https://doi.org/10.1007/s12188-011-0049-0
  10. 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
  11. S. Ianus and M. Visinescu, 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
  12. 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.
  13. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York, 1969.
  14. 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
  15. B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469. http://projecteuclid.org/euclid.mmj/1028999604 https://doi.org/10.1307/mmj/1028999604
  16. 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
  17. K.-S. Park, h-semi-invariant submersions, Taiwanese J. Math. 16 (2012), no. 5, 1865-1878. https://doi.org/10.11650/twjm/1500406802
  18. K.-S. Park, h-semi-slant submersions from almost quaternionic Hermitian manifolds, Taiwanese J. Math. 18 (2014), no. 6, 1909-1926. https://doi.org/10.11650/tjm.18.2014.4079
  19. 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
  20. 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
  21. B. Sahin, Invariant and anti-invariant Riemannian maps to Kahler manifolds, Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 3, 337-355. https://doi.org/10.1142/S0219887810004324
  22. B. Sahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102) (2011), no. 1, 93-105.
  23. 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
  24. B. Sahin, Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math. 17 (2013), no. 2, 629-659. https://doi.org/10.11650/tjm.17.2013.2191
  25. 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
  26. 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.