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New results toward the classification of biharmonic submanifolds in 𝕊n Cover

New results toward the classification of biharmonic submanifolds in 𝕊n

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică
Volume 20 (2012): Issue 2 (June 2012)
By: Cristina Flaut  
Open Access
|May 2013

References

  1. [1] A. Balmuş, S. Montaldo, C. Oniciuc. Biharmonic PNMC submanifolds in spheres, to appear in Ark. Mat.
    Search in Google Scholar
  2. [2] A. Balmuş, S. Montaldo, C. Oniciuc. Properties of biharmonic submani- folds in spheres. J. Geom. Symmetry Phys. 17 (2010), 87 102.
    Search in Google Scholar
  3. [3] A. Balmuş, S. Montaldo, C. Oniciuc. Biharmonic hypersurfaces in 4- dimensional space forms. Math. Nachr. 283 (2010), 1696-1705.
    Search in Google Scholar
  4. [4] A. Balmuş, S. Montaldo, C. Oniciuc. Classification results and new exam- ples of proper biharmonic submanifolds in spheres. Note Mat. 28 (2008), suppl. 1, 49-61.
    Search in Google Scholar
  5. [5] A. Balmuş, S. Montaldo, C. Oniciuc. Classification results for biharmonic submanifolds in spheres. Israel J. Math. 168 (2008), 201-220.
    Search in Google Scholar
  6. [6] A. Balmuş, C. Oniciuc. Biharmonic submanifolds with parallel mean cur- vature vector field in spheres. J. Math. Anal. Appi. 386 (2012), 619-630.
    Search in Google Scholar
  7. [7] A. Balmuş, C. Oniciuc. Biharmonic surfaces of §4. Kyushu .J. Math. 63 (2009), 339-345.10.2206/kyushujm.63.339
    Search in Google Scholar
  8. [8] A. Besse. Einstein manifolds. Springer-Verlag, Berlin, 1987.10.1007/978-3-540-74311-8
    Search in Google Scholar
  9. [9] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds in spheres. Israel J. Math. 130 (2002), 109-123.10.1007/BF02764073
    Search in Google Scholar
  10. [10] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds of §3. Internat. J. Math. 12 (2001), 807-876.
    Search in Google Scholar
  11. [11] S. Chang. A closed hypersurface with constant scalar and mean curvature in §4 is isoparametric. Comm. Anal. Geom 1 (1993), 71-100.10.4310/CAG.1993.v1.n1.a4
    Search in Google Scholar
  12. [12] B-Y. Chen. Pseudo-Riemannian geometry, invariants and applications, World Scientific, Hackensack, NJ, 2011.10.1142/8003
    Search in Google Scholar
  13. [13] B-Y. Chen. Complete| classification of parallel spatial surfaces in pseudo- Riemannian space forms with arbitrary index and dimension. J. Geom. Phys. 00 (2010), 200-280.10.1016/j.geomphys.2009.09.012
    Search in Google Scholar
  14. [14] B-Y. Chen. A report on submanifolds of finite type. Soochow J. Math. 22 (1990), 117-337.
    Search in Google Scholar
  15. [15] B-Y. Chen. Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17 (1991), 109-188.
    Search in Google Scholar
  16. [16] B-Y. Chen. Total Mean Curvature and Submanifolds of Finite Type. Se- ries in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.10.1142/0065
    Search in Google Scholar
  17. [17] J.H. Chen. Compact 2-harmonic hypersurfaces in §n+1(l). Acta Math. Sinica 30 (1993), 49-50.
    Search in Google Scholar
  18. [18] Q.-M. Cheng. Compact hypersurfaces in a unit sphere with infinite fun- damental group. Pacific .1. Math. 212 (2003), 341-347.10.2140/pjm.2003.212.49
    Search in Google Scholar
  19. [19] S.S. Chern, M. doCarmo, S. Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length. 1970 Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, 111., 1908), 59-75 Springer, New York.10.1007/978-3-642-48272-4_2
    Search in Google Scholar
  20. [20] F. Defever. Hypersurfaces of E4 with harmonic mean curvature vector. Math. Nachr. 190 (1998), 01-09.
    Search in Google Scholar
  21. [21] I. Dimitric. Submanifolds of Em with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica 20 (1992), 53-05.
    Search in Google Scholar
  22. [22] J. Erbacher. Reduction of the codimension of an isometric immersion. J. Differ. Geom. 5 (1971), 333-340.10.4310/jdg/1214429997
    Search in Google Scholar
  23. [23] D. Fetcu, C. Oniciuc. Biharmonic integral C-parallel submanifolds in 7- dimensional Sasakian space forms, Tohoku Math. J., to appear.
    Search in Google Scholar
  24. [24] T. Hasanis, T. Vlachos. Hypersurfaces in E4 with harmonic mean curva- ture vector field. Math. Nachr. 172 (1995), 145-169.
    Search in Google Scholar
  25. [25] T. Ichiyama, .1.1. Inoguchi, H. Urakawa. Bi-harmonic maps and bi-Yang- Mills fields. Note Mat. 28 suppl. n. 1 (2008), 233-275.
    Search in Google Scholar
  26. [26] G.Y. Jiang. 2-harmonic isometric immersions between Riemannian man- ifolds. Chinese Ann. Math. Ser. A 7 (1986), 130-144.
    Search in Google Scholar
  27. [27] H.B. Lawson. Local rigidity theorems for minimal hypersurfaces. Ann. of Math. (2) 89 (1969), 187-197.10.2307/1970816
    Search in Google Scholar
  28. [28] P. Li. Harmonic functions and applications to complete manifolds. XIV Escola de Geometria Diferencial (IMPA), Rio de Janeiro, 2006.
    Search in Google Scholar
  29. [29] N. Nakauchi, H. Urakawa. Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature. Ann. Glob. Anal. Geom. 40 (2011), 125-131.
    Search in Google Scholar
  30. [30] N. Nakauchi, H. Urakawa. Biharmonic submanifolds in a Riemannian manifold with non-positive curvature. Results Math., to appear.
    Search in Google Scholar
  31. [31] C. Oniciuc. Tangency and Harmonicity Properties. PhD The- sis, Geometry Balkan Press 2003, http://www.mathem.pub.ro/-dgds/mono/dgdsmono.htm
    Search in Google Scholar
  32. [32] C. Oniciuc. Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. ALL Cuza Iasi Mat (N.S.) 48 (2002), 237-248.
    Search in Google Scholar
  33. [33] T[ Sasahara. Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors. Pubi. Math. Debrecen 67 (2005), 285-303.10.1515/dema-2005-0121
    Search in Google Scholar
  34. [34] Y.-L. Ou, L. Tang. On the generalized Chen’s conjecture on biharmonic submanifolds. Michigan Math. J., to appear.
    Search in Google Scholar
  35. [35] C. Peng, C. Terng. Minimal hypersurfaces of spheres with constant scalar curvature. Seminar on minimal submanifolds, 177-198, Ann. of Math. Stud., 103, Princeton Univ. Press, Princeton, NJ, 1983.10.1515/9781400881437-010
    Search in Google Scholar
  36. [36] W. Santos. Submanifolds with parallel mean curvature vector in spheres. Tohoku Math. J. 46 (1994), 403-415.10.2748/tmj/1178225720
    Search in Google Scholar
  37. [37] J. Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. 88 (1968), 62-105.[38] X.F. Wang, L. Wu. Proper biharmonic submanifolds in a sphere. Acta Math. Sin. (Engl. Ser.) 28 (2012), 205-218.10.1007/s10114-012-9018-5
    Search in Google Scholar
  38. [39] S.-T. Yau. Harmonic functions on complete Riemannian manifolds. Com- mun. Pure Appi. Math. 28 (1975), 201-228.10.1002/cpa.3160280203
    Search in Google Scholar
  39. [40] S.-T. Yau. Submanifolds with constant mean curvature I. Amer. J. Math. 90 (1974), 34G-36G.
    Search in Google Scholar
  40. [41] W. Zhang. New examples of biharmonic submanifolds in CPn and S2n+1. An. Stiint. Univ. Al.I. Cuza Iasi Mat (N.S.) 57 (2011), 207-218. 10.2478/v10157-010-0046-0
    Search in Google Scholar
DOI: https://doi.org/10.2478/v10309-012-0043-2 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784
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Language: English
Page range: 89 - 114
Published on: May 17, 2013
Published by: Ovidius University of Constanta
In partnership with: Paradigm Publishing Services
Publication frequency: 3 issues per year
Keywords:
biharmonic submanifolds,
CMC submanifolds,
parallel submanifolds
Related subjects:
Mathematics,
General mathematics

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This work is licensed under the Creative Commons License.

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