Notes on critical almost Hermitian structures
Bull. Korean Math. Soc. 2010 Vol. 47, No. 1, 167-178
https://doi.org/10.4134/BKMS.2010.47.1.167
Printed January 1, 2010
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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