DOI QR코드

DOI QR Code

HARMONIC MAPS AND BIHARMONIC MAPS ON PRINCIPAL BUNDLES AND WARPED PRODUCTS

  • Received : 2017.04.17
  • Accepted : 2018.01.30
  • Published : 2018.05.01

Abstract

In this paper, we study harmonic maps and biharmonic maps on the principal G-bundle in Kobayashi and Nomizu and also the warped product $P=M{\times}_fF$ for a $C^{\infty}$(M) function f on M studied by Bishop and O'Neill, and Ejiri.

Keywords

Acknowledgement

Supported by : Japan Society for the Promotion of Science

References

  1. K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351-355. https://doi.org/10.1007/s10711-012-9778-1
  2. A. Balmus, S. Montaldo, and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201-220. https://doi.org/10.1007/s11856-008-1064-4
  3. A. Balmus, S. Montaldo, and C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010), no. 12, 1696-1705. https://doi.org/10.1002/mana.200710176
  4. 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
  5. C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
  6. R. Caddeo, S. Montaldo, and P. Piu, On biharmonic maps, in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 286-290, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2000.
  7. I. Castro, H. Li, and F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, Pacific J. Math. 227 (2006), no. 1, 43-63. https://doi.org/10.2140/pjm.2006.227.43
  8. B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169-188.
  9. F. Defever, Hypersurfaces of ${\mathbf{E}}^4$ with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61-69. https://doi.org/10.1002/mana.19981960104
  10. J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1983.
  11. N. Ejiri, A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z. 168 (1979), no. 3, 207-212. https://doi.org/10.1007/BF01214512
  12. D. Fetcu and C. Oniciuc, Biharmonic integral C-parallel submanifolds in 7-dimensional Sasakian space forms, Tohoku Math. J. (2) 64 (2012), no. 2, 195-222. https://doi.org/10.2748/tmj/1341249371
  13. Th. Hasanis and Th. Vlachos, Hypersurfaces in ${\mathbb{E}}^4$ with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145-169. https://doi.org/10.1002/mana.19951720112
  14. T. Ichiyama, J. Inoguchi, and H. Urakawa, Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009), suppl. 1, 233-275.
  15. T. Ichiyama, J. Inoguchi, and H. Urakawa, Classifications and isolation phenomena of bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 30 (2010), no. 2, 15-48.
  16. J.-I. Inoguchi, Submanifolds with harmonic mean curvature vector field in contact 3-manifolds, Colloq. Math. 100 (2004), no. 2, 163-179. https://doi.org/10.4064/cm100-2-2
  17. H. Iriyeh, Hamiltonian minimal Lagrangian cones in ${\mathbb{C}}^m$, Tokyo J. Math. 28 (2005), no. 1, 91-107. https://doi.org/10.3836/tjm/1244208282
  18. S. Ishihara and S. Ishikawa, Notes on relatively harmonic immersions, Hokkaido Math. J. 4 (1975), no. 2, 234-246. https://doi.org/10.14492/hokmj/1381758762
  19. G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389-402.
  20. T. Kajigaya, Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds, Tohoku Math. J. (2) 65 (2013), no. 4, 523-543. https://doi.org/10.2748/tmj/1386354294
  21. S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York, 1972.
  22. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York, 1963.
  23. N. Koiso and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold, accepted in Osaka J. Math.
  24. Y. Luo, Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, Results Math. 65 (2014), no. 1-2, 49-56. https://doi.org/10.1007/s00025-013-0328-4
  25. Y. Luo, On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys. 88 (2015), 76-87. https://doi.org/10.1016/j.geomphys.2014.11.004
  26. Y. Luo, Remarks on the nonexistence of biharmonic maps, Arch. Math. (Basel) 107 (2016), no. 2, 191-200. https://doi.org/10.1007/s00013-016-0924-0
  27. E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (2005), no. 3, 729-745. https://doi.org/10.1112/S0010437X04001204
  28. E. Loubeau and C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5239-5256. https://doi.org/10.1090/S0002-9947-07-03934-7
  29. E. Loubeau and Y.-L. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J. (2) 62 (2010), no. 1, 55-73. https://doi.org/10.2748/tmj/1270041027
  30. S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kahler manifolds, Glasg. Math. J. 55 (2013), no. 2, 465-480. https://doi.org/10.1017/S0017089512000730
  31. S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), no. 2, 1-22.
  32. Y. Nagatomo, Harmonic maps into Grassmannians and a generalization of do Carmo-Wallach theorem, in Riemann surfaces, harmonic maps and visualization, 41-52, OCAMI Stud., 3, Osaka Munic. Univ. Press, Osaka, 2008.
  33. H. Naito and H. Urakawa, Conformal change of Riemannian metrics and biharmonic maps, Indiana Univ. Math. J. 63 (2014), no. 6, 1631-1657. https://doi.org/10.1512/iumj.2014.63.5424
  34. N. Nakauchi and H. Urakawa, Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature, Ann. Global Anal. Geom. 40 (2011), no. 2, 125-131. https://doi.org/10.1007/s10455-011-9249-1
  35. N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013), no. 1-2, 467-474. https://doi.org/10.1007/s00025-011-0209-7
  36. N. Nakauchi, H. Urakawa, and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata 169 (2014), 263-272. https://doi.org/10.1007/s10711-013-9854-1
  37. S. Ohno, T. Sakai, and H. Urakawa, Rigidity of transversally biharmonic maps between foliated Riemannian manifolds, Hokkaido Math. J., 2017.
  38. C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 48 (2002), no. 2, 237-248.
  39. Y.-L. Ou and L. Tang, The generalized Chen's conjecture on biharmonic submanifolds is false, arXiv: 1006.1838v1.
  40. Y.-L. Ou and L. Tang, On the generalized Chen's conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), no. 3, 531-542. https://doi.org/10.1307/mmj/1347040257
  41. T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), no. 3-4, 285-303.
  42. T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms, Canad. Math. Bull. 51 (2008), no. 3, 448-459. https://doi.org/10.4153/CMB-2008-045-0
  43. T. Sasahara, A class of biminimal Legendrian submanifolds in Sasakian space forms, Math. Nachr. 287 (2014), no. 1, 79-90. https://doi.org/10.1002/mana.201200153
  44. R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229-236. https://doi.org/10.1090/S0002-9939-1975-0375386-2
  45. T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. https://doi.org/10.2969/jmsj/01840380
  46. K. Tsukada, Eigenvalues of the Laplacian of warped product, Tokyo J. Math. 3 (1980), no. 1, 131-136. https://doi.org/10.3836/tjm/1270216086
  47. H. Urakawa, Calculus of Variations and Harmonic Maps, translated from the 1990 Japanese original by the author, Translations of Mathematical Monographs, 132, American Mathematical Society, Providence, RI, 1993.
  48. H. Urakawa, CR rigidity of pseudo harmonic maps and pseudo biharmonic maps, Hokkaido Math. J. 46 (2017), no. 2, 141-187. https://doi.org/10.14492/hokmj/1498788016
  49. Z.-P. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011), no. 3-4, 917-925. https://doi.org/10.1007/s00209-010-0766-6