Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 253-270, 2017


A curvature identity on a 6-dimensional Riemannian manifold and its applications

Yunhee Euh, Jeong Hyeong Park, Kouei Sekigawa

Received October 9, 2015.  First published February 24, 2017.

Abstract:  We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold "a harmonic manifold is locally symmetric" and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting.
Keywords:  Chern-Gauss-Bonnet theorem; curvature identity; locally harmonic manifold
Classification MSC:  53B20, 53C25
DOI:  10.21136/CMJ.2017.0540-15


References:
[1] T. Arias-Marco, O. Kowalski: Classification of 4-dimensional homogeneous weakly Einstein manifolds. Czech. Math. J. 65 (2015), 21-59. DOI 10.1007/s10587-015-0159-4 | MR 3336024 | Zbl 06433720
[2] M. Berger: Quelques formules de variation pour une structure riemannienne. Ann. Sci. Éc. Norm. Supér French 3 (1970), 285-294. MR 0278238 | Zbl 0204.54802
[3] J. Berndt, F. Tricerri, L. Vanhecke: Generalized Heisenberg Groups and Damek Ricci Harmonic Spaces. Lecture Notes in Mathematics 1598, Springer, Berlin (1995). DOI 10.1007/BFb0076902 | MR 1340192 | Zbl 0818.53067
[4] A. L. Besse: Manifolds All of Whose Geodesics Are Closed. Ergebnisse der Mathematik und ihrer Grenzgebiete 93, Springer, Berlin (1978). DOI 10.1007/978-3-642-61876-5_3 | MR 496885 | Zbl 0387.53010
[5] E. Boeckx, L. Vanhecke: Unit tangent sphere bundles with constant scalar curvature. Czech. Math. J. 51 (2001), 523-544. DOI 10.1023/A:1013779805244 | MR 1851545 | Zbl 1079.53063
[6] P. Carpenter, A. Gray, T. J. Willmore: The curvature of Einstein symmetric spaces. Q. J. Math., Oxf. II. 33 (1982), 45-64. DOI 10.1093/qmath/33.1.45 | MR 0689850 | Zbl 0509.53045
[7] S. H. Chun, J. H. Park, K. Sekigawa: H-contact unit tangent sphere bundles of Einstein manifolds. Q. J. Math. 62 (2011), 59-69. DOI 10.1093/qmath/hap025 | MR 2774353 | Zbl 1222.53047
[8] E. T. Copson, H. S. Ruse: Harmonic Riemannian space. Proc. R. Soc. Edinb. 60 (1940), 117-133. DOI 10.1017/s0370164600020095 | MR 0002249 | Zbl 0027.26001
[9] E. Damek, F. Ricci: A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc., New. Ser. 27 (1992), 139-142. DOI 10.1090/S0273-0979-1992-00293-8 \goodbreak | MR 1142682 | Zbl 0755.53032
[10] Y. Euh, P. Gilkey, J. H. Park, K. Sekigawa: Transplanting geometrical structures. Differ. Geom. Appl. 31 (2013), 374-387. DOI 10.1016/j.difgeo.2013.03.006 | MR 3049632 | Zbl 1282.53024
[11] Y. Euh, J. H. Park, K. Sekigawa: A curvature identity on a 4-dimensional Riemannian manifold. Result. Math. 63 (2013), 107-114. DOI 10.1007/s00025-011-0164-3 | MR 3009674 | Zbl 1273.53009
[12] Y. Euh, J. H. Park, K. Sekigawa: A generalization of a 4-dimensional Einstein manifold. Math. Slovaca 63 (2013), 595-610. DOI 10.2478/s12175-013-0121-6 | MR 3071978 | Zbl 06201661
[13] P. Gilkey, J. H. Park, K. Sekigawa: Universal curvature identities. Differ. Geom. Appl. 29 (2011), 770-778. DOI 10.1016/j.difgeo.2011.08.005 | MR 2846274 | Zbl 1259.53013
[14] P. Gilkey, J. H. Park, K. Sekigawa: Universal curvature identities II. J. Geom. Phys. 62 (2012), 814-825. DOI 10.1016/j.geomphys.2012.01.002 | MR 2888984 | Zbl 1246.53022
[15] P. Gilkey, J. H. Park, K. Sekigawa: Universal curvature identities III. Int. J. Geom. Methods Mod. Phys. 10 Article ID 1350025, 21 pages (2013). DOI 10.1142/S0219887813500254 | MR 3056523 | Zbl 06200755
[16] P. Gilkey, J. H. Park, K. Sekigawa: Universal curvature identities and Euler Lagrange formulas for Kähler manifolds. J. Math. Soc. Japan 68 (2016), 459-487. DOI 10.2969/jmsj/06820459 | MR 3488133 | Zbl 06597343
[17] A. Gray, T. J. Willmore: Mean-value theorems for Riemannian manifolds. Proc. R. Soc. Edinb., Sect. A 92 (1982), 343-364. DOI 10.1017/S0308210500032571 | MR 0677493 | Zbl 0495.53040
[18] P. Kreyssig: An introduction to harmonic manifolds and the Lichnerowicz conjecture. Available at arXiv:1007.0477v1.
[19] A. J. Ledger: Harmonic Spaces. Ph.D. Thesis, University of Durham, Durham (1954).
[20] A. J. Ledger: Symmetric harmonic spaces. J. London Math. Soc. 32 (1957), 53-56. DOI 10.1112/jlms/s1-32.1.53 | MR 0083796 | Zbl 0084.37406
[21] A. Lichnerowicz: Sur les espaces riemanniens complétement harmoniques. Bull. Soc. Math. Fr. French 72 (1944), 146-168. MR 0012886 | Zbl 0060.38506
[22] A. Lichnerowicz: Géométrie des groupes de transformations. Travaux et recherches mathématiques 3, Dunod, Paris French (1958). MR 0124009 | Zbl 0096.16001
[23] Y. Nikolayevsky: Two theorems on harmonic manifolds. Comment. Math. Helv. 80 (2005), 29-50. DOI 10.4171/CMH/2 | MR 2130564 | Zbl 1078.53032
[24] E. M. Patterson: A class of critical Riemannian metrics. J. Lond. Math. Soc., II. Ser. 23 (1981), 349-358. DOI 10.1112/jlms/s2-23.2.349 | MR 0609115 | Zbl 0417.53025
[25] T. Sakai: On eigen-values of Laplacian and curvature of Riemannian manifold. Tohoku Math. J., II. Ser. 23 (1971), 589-603. DOI 10.2748/tmj/1178242547 | MR 0303465 | Zbl 0237.53040
[26] K. Sekigawa: On 4-dimensional connected Einstein spaces satisfying the condition $R(X,Y)\cdot R=0$. Sci. Rep. Niigata Univ., Ser. A 7 (1969), 29-31. MR 0261490 | Zbl 0345.53035
[27] K. Sekigawa, L. Vanhecke: Volume-preserving geodesic symmetries on four-dimensional Kähler manifolds. Proc. Symp. Differential geometry, Pe niscola 1985, Lect. Notes Math. 1209, Springer, Berlin 275-291 (1986). DOI 10.1007/BFb0076638 | MR 0863763 | Zbl 0605.53031
[28] Z. I. Szabó: The Lichnerowicz conjecture on harmonic manifolds. J. Differ. Geom. 31 (1990), 1-28. MR 1030663 | Zbl 0686.53042
[29] S. Tachibana: On the characteristic function of spaces of constant holomorphic curvature. Colloq. Math. 26 (1972), 149-155. MR 0336678 | Zbl 0223.53044
[30] A. G. Walker: On Lichnerowicz's conjecture for harmonic 4-spaces. J. Lond. Math. Soc. 24 (1949), 21-28. DOI 10.1112/jlms/s1-24.1.21 | MR 0030280 | Zbl 0032.18801
[31] Y. Watanabe: On the characteristic function of harmonic Kählerian spaces. Tohoku Math. J., II. Ser. 27 (1975), 13-24. DOI 10.2748/tmj/1178241030 | MR 0365439 | Zbl 0311.53068
[32] Y. Watanabe: On the characteristic functions of harmonic quaternion Kählerian spaces. Kodai Math. Semin. Rep. 27 (1976), 410-420. DOI 10.2996/kmj/1138847322 | MR 0418004 | Zbl 0336.53019
[33] Y. Watanabe: The sectional curvature of a 5-dimensional harmonic Riemannian manifold. Kodai Math. J. 6 (1983), 106-109. DOI 10.2996/kmj/1138036667 \filbreak | MR 0698331 | Zbl 0519.53013

Affiliations:   Yunhee Euh, Department of Mathematical Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea e-mail: yheuh@snu.ac.kr; Jeong Hyeong Park, Department of Mathematics, Sungkyunkwan University, 2066, Seobu-ro, Jangan-gu, Suwon 16419, Gyeong Gi-Do, Korea, e-mail: parkj@skku.edu; Kouei Sekigawa, Department of Mathematics, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan, e-mail: sekigawa@math.sc.niigata-u.ac.jp

Springer subscribers can access the papers on Springer website.
Access to full texts on this site is restricted to subscribers of Myris Trade. To activate your access, please send an e-mail to myris@myris.cz.
[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

 
PDF available at: