본문 바로가기
KERD
논문KCI 등재

Harmonic maps and biharmonic maps on principal bundles and warped products

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

주제분류수학

초록

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

Abstract

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

참고문헌 (49)

  1. K. Akutagawa, "Biharmonic properly immersed submanifolds in Euclidean spaces", Geom. Dedicata, 164, pp.351~355 (2013)[DOI]
  2. A. Balmus, "Classification results for biharmonic submanifolds in spheres", Israel J. Math., 168, pp.201~220 (2008)[DOI]
  3. A. Balmus, "Biharmonic hypersurfaces in 4-dimensional space forms", Math. Nachr., 283(12), pp.1696~1705 (2010)[DOI]
  4. R. L. Bishop, "Manifolds of negative curvature", Trans. Amer. Math. Soc., 145, pp.1~49 (1969)[DOI]
  5. C. P. Boyer, "Sasakian Geometry, Oxford Mathematical Monographs" (2008), Oxford University Press
  6. R. Caddeo, "On biharmonic maps, in Global differential geometry: the mathematical legacy of Alfred Gray", pp.286~290 (2000), Amer. Math. Soc
  7. I. Castro, "Hamiltonian-minimal Lagrangian submanifolds in complex space forms", Pacific J. Math., 227(1), pp.43~63 (2006)[DOI]
  8. B. Y. Chen, "Some open problems and conjectures on submanifolds of finite type", Soochow J. Math., 17(2), pp.169~188 (1991)
  9. F. Defever, "Hypersurfaces of <TEX>${\mathbf{E}}^4$<", TEX> with harmonic mean curvature vector, 1998, pp.196, 61~69
  10. J. Eells, "Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, 50" (1983), Published for the Conference Board of the Mathematical Sciences
  11. N. Ejiri, "A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric", Math. Z., 168(3), pp.207~212 (1979)[DOI]
  12. D. Fetcu, "Biharmonic integral C-parallel submanifolds in 7-dimensional Sasakian space forms", Tohoku Math. J., 64(2), pp.195~222 (2012)[DOI]
  13. Th. Hasanis, "Hypersurfaces in <TEX>${\mathbb{E}}^4$<", TEX> with harmonic mean curvature vector field, 1995, pp.172, 145~169
  14. T. Ichiyama, "Bi-harmonic maps and bi-Yang-Mills fields", Note Mat., 28(suppl. 1), pp.233~275 (2009)
  15. T. Ichiyama, "Classifications and isolation phenomena of bi-harmonic maps and bi-Yang-Mills fields", Note Mat., 30(2), pp.15~48 (2010)
  16. J.-I. Inoguchi, "Submanifolds with harmonic mean curvature vector field in contact 3-manifolds", Colloq. Math., 100(2), pp.163~179 (2004)[DOI]
  17. H. Iriyeh, "Hamiltonian minimal Lagrangian cones in <TEX>${\mathbb{C}}^m$<", TEX>, 2005, pp.28(1), 91~107
  18. S. Ishihara, "Notes on relatively harmonic immersions", Hokkaido Math. J., 4(2), pp.234~246 (1975)[DOI]
  19. G. Y. Jiang, "2-harmonic maps and their first and second variational formulas", Chinese Ann. Math. Ser. A, 7(4), pp.389~402 (1986)
  20. T. Kajigaya, "Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds", Tohoku Math. J., 65(4), pp.523~543 (2013)[DOI]
  21. S. Kobayashi, "Transformation Groups in Differential Geometry" (1972), Springer-Verlag
  22. S. Kobayashi, "Foundations of Differential Geometry. Vol I" (1963), Interscience Publishers, a division of John Wiley & Sons
  23. N. Koiso, "Biharmonic submanifolds in a Riemannian manifold", Osaka J. Math.
  24. Y. Luo, "Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal", Results Math., 65(1-2), pp.49~56 (2014)[DOI]
  25. Y. Luo, "On biharmonic submanifolds in non-positively curved manifolds", J. Geom. Phys., 88, pp.76~87 (2015)[DOI]
  26. Y. Luo, "Remarks on the nonexistence of biharmonic maps", Arch. Math. (Basel), 107(2), pp.191~200 (2016)[DOI]
  27. E. Loubeau, "The index of biharmonic maps in spheres", Compos. Math., 141(3), pp.729~745 (2005)[DOI]
  28. E. Loubeau, "On the biharmonic and harmonic indices of the Hopf map", Trans. Amer. Math. Soc., 359(11), pp.5239~5256 (2007)[DOI]
  29. E. Loubeau, "Biharmonic maps and morphisms from conformal mappings", Tohoku Math. J., 62(1), pp.55~73 (2010)[DOI]
  30. S. Maeta, "Biharmonic Lagrangian submanifolds in Kahler manifolds", Glasg. Math. J., 55(2), pp.465~480 (2013)[DOI]
  31. S. Montaldo, "A short survey on biharmonic maps between Riemannian manifolds", Rev. Un. Mat. Argentina, 47(2), pp.1~22 (2006)
  32. Y. Nagatomo, "Riemann surfaces, harmonic maps and visualization", pp.41~52 (2008), Osaka Munic. Univ. Press
  33. H. Naito, "Conformal change of Riemannian metrics and biharmonic maps", Indiana Univ. Math. J., 63(6), pp.1631~1657 (2014)[DOI]
  34. N. Nakauchi, "Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature", Ann. Global Anal. Geom., 40(2), pp.125~131 (2011)[DOI]
  35. N. Nakauchi, "Biharmonic submanifolds in a Riemannian manifold with non-positive curvature", Results Math., 63(1-2), pp.467~474 (2013)[DOI]
  36. N. Nakauchi, "Biharmonic maps into a Riemannian manifold of non-positive curvature", Geom. Dedicata, 169, pp.263~272 (2014)[DOI]
  37. S. Ohno, "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(2), pp.237~248 (2002)
  39. Y.-L. Ou, "The generalized Chen's conjecture on biharmonic submanifolds is false", arXiv: 1006.1838v1, ~
  40. Y.-L. Ou, "On the generalized Chen's conjecture on biharmonic submanifolds", Michigan Math. J., 61(3), pp.531~542 (2012)[DOI]
  41. T. Sasahara, "Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors", Publ. Math. Debrecen, 67(3-4), pp.285~303 (2005)
  42. T. Sasahara, "Stability of biharmonic Legendrian submanifolds in Sasakian space forms", Canad. Math. Bull., 51(3), pp.448~459 (2008)[DOI]
  43. T. Sasahara, "A class of biminimal Legendrian submanifolds in Sasakian space forms", Math. Nachr., 287(1), pp.79~90 (2014)[DOI]
  44. R. T. Smith, "The second variation formula for harmonic mappings", Proc. Amer. Math. Soc., 47, pp.229~236 (1975)[DOI]
  45. T. Takahashi, "Minimal immersions of Riemannian manifolds", J. Math. Soc. Japan, 18, pp.380~385 (1966)[DOI]
  46. K. Tsukada, "Eigenvalues of the Laplacian of warped product", Tokyo J. Math., 3(1), pp.131~136 (1980)[DOI]
  47. H. Urakawa, "Calculus of Variations and Harmonic Maps, translated from the 1990 Japanese original by the author, Translations of Mathematical Monographs, 132" (1993)
  48. H. Urakawa, "CR rigidity of pseudo harmonic maps and pseudo biharmonic maps", Hokkaido Math. J., 46(2), pp.141~187 (2017)[DOI]
  49. Z.-P. Wang, "Biharmonic Riemannian submersions from 3-manifolds", Math. Z., 269(3-4), pp.917~925 (2011)[DOI]

관련 논문AI추천

이 간행물의 다른 논문 (12건)