DOI QR코드

DOI QR Code

Biharmonic Submanifolds of Quaternionic Space Forms

  • Kacimi, Bouazza (Department of Mathematics, Faculty of Exact Sciences, Mascara University) ;
  • Cherif, Ahmed Mohammed (Department of Mathematics, Faculty of Exact Sciences, Mascara University)
  • 투고 : 2018.08.01
  • 심사 : 2019.01.28
  • 발행 : 2019.12.23

초록

In this paper, we consider biharmonic submanifolds of a quaternionic space form. We give the necessary and sufficient conditions for a submanifold to be biharmonic in a quaternionic space form, we study different particular cases for which we obtain some non-existence results and curvature estimates.

키워드

참고문헌

  1. P. Baird and J. C. Wood, Harmonic morphisms between Riemannain manifolds, Clarendon Press Oxford, 2003.
  2. J. Berndt, Real hypersurfaces in quaternionic space space forms, J. reine Angew. Math., 419(1991), 9-26.
  3. R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of ${\mathbb{S}^3}$, Internat. J. Math., 12(2001), 867-876. https://doi.org/10.1142/S0129167X01001027
  4. R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math., 130(2002), 109-123. https://doi.org/10.1007/BF02764073
  5. B.-Y. Chen, Totally umbilical submanifolds of quaternion-space-forms, J. Austral. Math. Soc. Ser. A, 26(1978), 154-162. https://doi.org/10.1017/S1446788700011642
  6. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86(1964), 109-160. https://doi.org/10.2307/2373037
  7. D. Fetcu, E. Loubeau, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of ${\mathbb{C}}P^n$, Math. Z., 266(2010), 505-531. https://doi.org/10.1007/s00209-009-0582-z
  8. D. Fetcu and C. Oniciuc, Explicit formulas for biharmonic submanifolds in Sasakian space forms, Pac. J. Math., 240(1)(2009), 85-107. https://doi.org/10.2140/pjm.2009.240.85
  9. T. Ichiyama, J. Inoguchi and H. Urakawa, Classifications and isolation phenomena of bi-harmonic maps and bi-Yang-Mills fields, Note Mat., 30(2)(2010), 15-48.
  10. S. Ishihara, Quaternion Kaehlerian manifolds, J. Differential Geometry, 9(1974), 483-500. https://doi.org/10.4310/jdg/1214432544
  11. G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A, 7(4)(1986), 389-402.
  12. H. S. Kim and J. S. Pak, Scalar curvature of QR-submanifolds with maximal QR-dimension in a quaternionic projective space, Indian J. Pure Appl. Math., 42(2)(2011), 109-126. https://doi.org/10.1007/s13226-011-0007-7
  13. J. S. Pak and T. H. Kang, Planar geodesic submanifolds in a quaternionic projective space, Geom. Dedicata, 26(1988), 139-155. https://doi.org/10.1007/BF00151666
  14. J. Roth, A note on biharmonic submanifolds of product spaces, J. Geom., 104(2013), 375-381. https://doi.org/10.1007/s00022-013-0168-0
  15. J. Roth and A. Upadhyay, Biharmonic submanifolds of generalized space forms, Diff. Geom. Appl., 50(2017), 88-104. https://doi.org/10.1016/j.difgeo.2016.11.003
  16. B. Sahin, Slant submanifolds of quaternion kaehler manifolds, Commun. Korean Math. Soc., 22(1)(2007), 123-135. https://doi.org/10.4134/CKMS.2007.22.1.123