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.

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

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 2010): 53B20, 53C25

DOI: 10.21136/CMJ.2017.0540-15

Full text available as PDF.


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