Biharmonic maps on principal \(G\)-bundles over complete Riemannian manifolds of nonpositive Ricci curvature. (English) Zbl 1472.58013
This article considers principal \(G\)-bundles, equipped with a Sasaki-type metric, over a Riemannian manifold and the canonical projection \(\pi\), which is then a Riemannian submersion.
The problem investigated is to find conditions such that \(\pi\) biharmonic implies \(\pi\) harmonic.
The first theorem proved is that if the principal \(G\)-bundle is compact and has non-positive Ricci curvature then \(\pi\) biharmonic implies \(\pi\) harmonic. The reader will notice in the proof that the one-form \(\alpha\) defined by Equation (3.7) is not quite well-defined on the base manifold unless the tension field of \(\pi\) is actually basic. This should then be compared with [C. Oniciuc, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 48, No. 2, 237–248 (2002; Zbl 1061.58015)].
The second set of conditions is to assume the principal \(G\)-bundle is non-compact complete with non-positive Ricci curvature and the energy and bienergy of \(\pi\) are finite. Then \(\pi\) biharmonic implies \(\pi\) harmonic. The proof of this second theorem is really only a rehash of N. Nakauchi et al. [Geom. Dedicata 169, 263–272 (2014; Zbl 1316.58012)].
The problem investigated is to find conditions such that \(\pi\) biharmonic implies \(\pi\) harmonic.
The first theorem proved is that if the principal \(G\)-bundle is compact and has non-positive Ricci curvature then \(\pi\) biharmonic implies \(\pi\) harmonic. The reader will notice in the proof that the one-form \(\alpha\) defined by Equation (3.7) is not quite well-defined on the base manifold unless the tension field of \(\pi\) is actually basic. This should then be compared with [C. Oniciuc, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 48, No. 2, 237–248 (2002; Zbl 1061.58015)].
The second set of conditions is to assume the principal \(G\)-bundle is non-compact complete with non-positive Ricci curvature and the energy and bienergy of \(\pi\) are finite. Then \(\pi\) biharmonic implies \(\pi\) harmonic. The proof of this second theorem is really only a rehash of N. Nakauchi et al. [Geom. Dedicata 169, 263–272 (2014; Zbl 1316.58012)].
Reviewer: Eric Loubeau (Brest)
References:
| [1] | K. Akutagawa and S. Maeta, Complete biharmonic submanifolds in the Euclidean spaces, Geom. Dedicata 164 (2013), 351-355. · Zbl 1268.53068 · doi: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. · Zbl 1172.58004 · doi: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), 1696-1705. · Zbl 1210.58013 · doi:10.1002/mana.200710176 |
| [4] | C. Boyer and K. Galicki, Sasakian geometry, Oxford Sci. Publ., 2008. · Zbl 1155.53002 |
| [5] | R. Caddeo, S. Montaldo, and P. Piu, On biharmonic maps, Contemp. Math. 288 (2001), 286-290. · Zbl 1010.58009 |
| [6] | I. Castro, H. Z. Li, and F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, Pacific J. Math. 227 (2006), 43-63. · Zbl 1129.53039 · doi:10.2140/pjm.2006.227.43 |
| [7] | B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), 169-188. · Zbl 0749.53037 |
| [8] | F. Defever, Hypersurfaces in \({\mathbb{E}}^4\) with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61-69. · Zbl 0944.53005 |
| [9] | J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Reg. Conf. Ser. Math., 50, Amer. Math. Soc., 1983. · Zbl 0515.58011 |
| [10] | D. Fetcu and C. Oniciuc, Biharmonic integral \({\mathcal{C}} \)-parallel submanifolds in 7-dimensional Sasakian space forms, Tohoku Math. J. 64 (2012), 195-222. · Zbl 1258.53059 · doi:10.2748/tmj/1341249371 |
| [11] | T. Hasanis and T. Vlachos, Hypersurfaces in \({\mathbb{E}}^4\) with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145-169. · Zbl 0839.53007 |
| [12] | T. Ichiyama, J. Inoguchi, and H. Urakawa, Biharmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009), 233-275. · Zbl 1201.58012 |
| [13] | T. Ichiyama, J. Inoguchi, and H. Urakawa, Classifications and isolation phenomena of biharmonic maps and bi-Yang-Mills fields, Note Mat. 30 (2010), 15-48. · Zbl 1244.58006 |
| [14] | J. Inoguchi, Submanifolds with harmonic mean curvature vector filed in contact 3-manifolds, Colloq. Math. 100 (2004), 163-179. · Zbl 1076.53065 · doi:10.4064/cm100-2-2 |
| [15] | H. Iriyeh, Hamiltonian minimal Lagrangian cones in \({\mathbb{C}}^m \), Tokyo J. Math. 28 (2005), 91-107. · Zbl 1087.53057 |
| [16] | S. Ishihara and S. Ishikawa, Notes on relatively harmonic immersions, Hokkaido Math. J. 4 (1975), 234-246. · Zbl 0311.53063 · doi:10.14492/hokmj/1381758762 |
| [17] | G. Y. Jiang, 2-harmonic maps and their first and second variational formula, Chinese Ann. Math. Ser. A 7 (1986), 388-402; Note Mat. 28 (2009), 209-232. · Zbl 1200.58015 |
| [18] | T. Kajigaya, Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds, Differential Geom. Appl. 37 (2014), 89-108. · Zbl 1303.53072 · doi:10.1016/j.difgeo.2014.09.004 |
| [19] | S. Kobayashi, Transformation groups in differential geometry, Springer, 1972. · Zbl 0246.53031 |
| [20] | E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (2005), 729-745. · Zbl 1075.58014 · doi:10.1112/S0010437X04001204 |
| [21] | E. Loubeau and C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc. 359 (2007), 5239-5256. · Zbl 1124.58009 · doi:10.1090/S0002-9947-07-03934-7 |
| [22] | E. Loubeau and Y.-L. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J. 62 (2010), 55-73. · Zbl 1202.53061 · doi:10.2748/tmj/1270041027 |
| [23] | S. Maeta and U. Urakawa, Biharmonic Lagrangian submanifolds in Kähler manifolds, Glasg. Math. J. 55 (2013), 465-480. · Zbl 1281.58009 · doi:10.1017/S0017089512000730 |
| [24] | S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), 1-22. · Zbl 1140.58004 |
| [25] | Y. Nagatomo, Harmonic maps into Grassmannians and a generalization of do Carmo-Wallach theorem, Proc. the 16th OCU intern. academic symp. 2008, OCAMI Stud., 3, pp. 41-52, 2008. · Zbl 1254.58007 |
| [26] | N. Nakauchi and H. Urakawa, Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature, Ann. Global Anal. Geom. 40 (2011), 125-131. · Zbl 1222.58010 · doi:10.1007/s10455-011-9249-1 |
| [27] | N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013), 467-474. · Zbl 1261.58011 · doi:10.1007/s00025-011-0209-7 |
| [28] | N. Nakauchi, H. Urakawa, and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata 169 (2014), 263-272. · Zbl 1316.58012 · doi:10.1007/s10711-013-9854-1 |
| [29] | S. Ohno, T. Sakai, and H. Urakawa, Biharmonic homogeneous hypersurfaces in compact symmetric spaces, Differential Geom. Appl. 43 (2015), 155-179. · Zbl 1336.58009 · doi:10.1016/j.difgeo.2015.09.005 |
| [30] | S. Ohno, T. Sakai, and H. Urakawa, Rigidity of transversally biharmonic maps between foliated Riemannian manifolds, Hokkaido Math. J. 47 (2018), 637-654. · Zbl 1408.58013 · doi:10.14492/hokmj/1537948835 |
| [31] | C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 48 (2002), no. 2, 237-248. · Zbl 1061.58015 |
| [32] | Y.-L. Ou and L. Tang, On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), 531-542. · Zbl 1268.58015 |
| [33] | Y.-L. Ou and L. Tang, The generalized Chen’s conjecture on biharmonic submanifolds is false, arXiv:1006.1838v1. |
| [34] | P. Baird and D. Kamissoko, On constructing biharmonic maps and metrics, Ann. Global Anal. Geom. 23 (2003), 65-75. · Zbl 1027.31004 · doi:10.1023/A:1021213930520 |
| [35] | T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), 285-303. · Zbl 1082.53067 |
| [36] | T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms, Canad. Math. Bull. 51 (2008), 448-459. · Zbl 1147.53315 · doi:10.4153/CMB-2008-045-0 |
| [37] | T. Sasahara, A class of biminimal Legendrian submanifolds in Sasaki space forms, Math. Nachr. 287 (2014), 79-90. · Zbl 1296.53128 · doi:10.1002/mana.201200153 |
| [38] | T. Talahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. · Zbl 0145.18601 |
| [39] | S. Ohno, T. Sakai, and H. Urakawa, CR rigidity of pseudo harmonic maps and pseudo biharmonic maps, Hokkaido Math. J. 47 (2018), 637-654. · Zbl 1408.58013 · doi:10.14492/hokmj/1537948835 |
| [40] | Z.-P. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011), 917-925. · Zbl 1235.53065 · doi:10.1007/s00209-010-0766-6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
