DOI QR코드

DOI QR Code

HOMOLOGY OF CONTACT CR-WARPED PRODUCT SUBMANIFOLDS OF AN ODD-DIMENSIONAL UNIT SPHERE

  • Received : 2013.12.17
  • Published : 2015.01.31

Abstract

We show that homology group on a contact CR-warped product submanifold in odd dimensional sphere is zero under certain conditions in terms of warping function and the dimension of the submanifold.

Keywords

References

  1. A. Bejancu, Geometry of CR-Submanifolds, Kluwer Academic Publishers, Dortrecht, 1986.
  2. R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
  3. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhauser, 2002.
  4. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifold, Monatsh. Math. 133 (2001), no. 3, 177-195. https://doi.org/10.1007/s006050170019
  5. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II, Monatsh. Math. 134 (2001), no. 2, 103-119. https://doi.org/10.1007/s006050170002
  6. B. Y. Chen, CR-warped products in complex projective spaces with compact holomorphic factor, Monatsh. Math. 141 (2004), no. 3, 177-186. https://doi.org/10.1007/s00605-002-0009-y
  7. B. Y. Chen, Geometry of warped product submanifolds: A survey, J. Adv. Math. Stud. 6 (2013), no. 2, 1-43.
  8. H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, Lecture Notes, Duke University, 2013.
  9. H. Federer and W. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458-520. https://doi.org/10.2307/1970227
  10. I. Hasegawa and I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata 102 (2003), 143-150. https://doi.org/10.1023/B:GEOM.0000006582.29685.22
  11. H. S. Kim and J. S. Pak, Sectional curvature of contact CR-submanifolds of an odd-dimensional unit sphere, Bull. Korean Math. Soc. 42 (2005), no. 4, 777-787. https://doi.org/10.4134/BKMS.2005.42.4.777
  12. H. S. Kim and J. S. Pak, Certain contact CR-submanifolds of an odd-dimensional unit sphere, Bull. Korean Math. Soc. 44 (2007), no. 1, 109-116. https://doi.org/10.4134/BKMS.2007.44.1.109
  13. H. S. Kim and J. S. Pak, Certain class of contact CR-submanifolds of an odd-dimensional unit sphere, Taiwanese J. Math. 14 (2010), no. 2, 629-646. https://doi.org/10.11650/twjm/1500405810
  14. H. S. Kim and J. S. Pak, Scalar curvature of contact CR-submanifolds in an odd-dimensional unit sphere, Bull. Korean Math. Soc. 47 (2010), no. 3, 541-549. https://doi.org/10.4134/BKMS.2010.47.3.541
  15. H. B. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. 98 (1973), 427-450. https://doi.org/10.2307/1970913
  16. P. F. Leung, On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold, Proc. Edinburgh Math. Soc. 28 (1985), no. 3, 305-311. https://doi.org/10.1017/S0013091500017119
  17. J. Liu and Q. Zhang, Non-existence of stable currents in submanifolds of the Euclidean spaces, J. Geom. 96 (2009), no. 1-2, 125-133. https://doi.org/10.1007/s00022-010-0024-4
  18. J. S. Pak, J.-H. Kwon, H. S. Kim, and Y.-M. Kim, Contact CR-submanifolds of an odd-dimensional unit sphere, Geom. Dedicata 114 (2005), 1-11. https://doi.org/10.1007/s10711-004-8175-9
  19. Y. L. Xin, An application of integral currents to the vanishing theorems, Sci. Sinica Ser. A 27 (1984), no. 3, 233-241.
  20. K. Yano and M. Kon, Structures on Manifolds, World Scientific, 1984.
  21. X. S. Zhang, Nonexistence of stable currents in submanifolds of a product of two spheres, Bull. Austral. Math. Soc. 44 (1991), no. 2, 325-336. https://doi.org/10.1017/S0004972700029762