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http://dx.doi.org/10.4134/JKMS.j170251

HARMONIC MAPS AND BIHARMONIC MAPS ON PRINCIPAL BUNDLES AND WARPED PRODUCTS  

Urakawa, Hajime (Global Learning Center Tohoku University)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.3, 2018 , pp. 553-574 More about this Journal
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
principal G bundle; Ricci curvature; harmonic map; biharmonic map; warped product;
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1 T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), no. 3-4, 285-303.
2 T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms, Canad. Math. Bull. 51 (2008), no. 3, 448-459.   DOI
3 T. Sasahara, A class of biminimal Legendrian submanifolds in Sasakian space forms, Math. Nachr. 287 (2014), no. 1, 79-90.   DOI
4 R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229-236.   DOI
5 T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385.   DOI
6 K. Tsukada, Eigenvalues of the Laplacian of warped product, Tokyo J. Math. 3 (1980), no. 1, 131-136.   DOI
7 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.
8 H. Urakawa, CR rigidity of pseudo harmonic maps and pseudo biharmonic maps, Hokkaido Math. J. 46 (2017), no. 2, 141-187.   DOI
9 Z.-P. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011), no. 3-4, 917-925.   DOI
10 Y. Luo, Remarks on the nonexistence of biharmonic maps, Arch. Math. (Basel) 107 (2016), no. 2, 191-200.   DOI
11 E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (2005), no. 3, 729-745.   DOI
12 S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), no. 2, 1-22.
13 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.   DOI
14 E. Loubeau and Y.-L. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J. (2) 62 (2010), no. 1, 55-73.   DOI
15 S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kahler manifolds, Glasg. Math. J. 55 (2013), no. 2, 465-480.   DOI
16 N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013), no. 1-2, 467-474.   DOI
17 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.
18 H. Naito and H. Urakawa, Conformal change of Riemannian metrics and biharmonic maps, Indiana Univ. Math. J. 63 (2014), no. 6, 1631-1657.   DOI
19 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.   DOI
20 N. Nakauchi, H. Urakawa, and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata 169 (2014), 263-272.   DOI
21 S. Ohno, T. Sakai, and H. Urakawa, Rigidity of transversally biharmonic maps between foliated Riemannian manifolds, Hokkaido Math. J., 2017.
22 N. Ejiri, A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z. 168 (1979), no. 3, 207-212.   DOI
23 C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 48 (2002), no. 2, 237-248.
24 Y.-L. Ou and L. Tang, The generalized Chen's conjecture on biharmonic submanifolds is false, arXiv: 1006.1838v1.
25 Y.-L. Ou and L. Tang, On the generalized Chen's conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), no. 3, 531-542.   DOI
26 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.   DOI
27 J.-I. Inoguchi, Submanifolds with harmonic mean curvature vector field in contact 3-manifolds, Colloq. Math. 100 (2004), no. 2, 163-179.   DOI
28 Th. Hasanis and Th. Vlachos, Hypersurfaces in ${\mathbb{E}}^4$ with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145-169.   DOI
29 T. Ichiyama, J. Inoguchi, and H. Urakawa, Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009), suppl. 1, 233-275.
30 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.
31 H. Iriyeh, Hamiltonian minimal Lagrangian cones in ${\mathbb{C}}^m$, Tokyo J. Math. 28 (2005), no. 1, 91-107.   DOI
32 S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York, 1972.
33 S. Ishihara and S. Ishikawa, Notes on relatively harmonic immersions, Hokkaido Math. J. 4 (1975), no. 2, 234-246.   DOI
34 G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389-402.
35 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.   DOI
36 S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York, 1963.
37 N. Koiso and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold, accepted in Osaka J. Math.
38 Y. Luo, Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, Results Math. 65 (2014), no. 1-2, 49-56.   DOI
39 K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351-355.   DOI
40 Y. Luo, On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys. 88 (2015), 76-87.   DOI
41 A. Balmus, S. Montaldo, and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201-220.   DOI
42 A. Balmus, S. Montaldo, and C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010), no. 12, 1696-1705.   DOI
43 I. Castro, H. Li, and F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, Pacific J. Math. 227 (2006), no. 1, 43-63.   DOI
44 R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49.   DOI
45 C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
46 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.
47 B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169-188.
48 F. Defever, Hypersurfaces of ${\mathbf{E}}^4$ with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61-69.   DOI
49 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.