Bulletin of the
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Bull. Korean Math. Soc. 2010; 47(1): 167-178

Printed January 1, 2010

https://doi.org/10.4134/BKMS.2010.47.1.167

Copyright © The Korean Mathematical Society.

Notes on critical almost Hermitian structures

Jung Chan Lee, Jeong Hyeong Park, and Kouei Sekigawa

Sungkyunkwan University, Sungkyunkwan University, and Niigata University

Abstract

We discuss the critical points of the functional $\mathcal {F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* ) dv_g$ on the spaces of all almost Hermitian structures $\mathcal{AH}(M)$ with ${(\lambda, \mu)} \in \mathbb{R}^2 - (0,0)$, where $\tau$ and $\tau^*$ being the scalar curvature and the $*$-scalar curvature of $(J, g)$, respectively. We shall give several characterizations of K\"{a}hler structure for some special classes of almost Hermitian manifolds, in terms of the critical points of the functionals $\mathcal {F}_{\lambda, \mu} (J, g)$ on $\mathcal{AH}(M)$. Further, we provide the almost Hermitian analogy of the Hilbert's result.

Keywords: critical almost Hermitian structure, Einstein-Hilbert functional

MSC numbers: 53C15, 53C55

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