# NOTES ON CRITICAL ALMOST HERMITIAN STRUCTURES

• Published : 2010.01.31
• 38 5

#### Abstract

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

#### Keywords

critical almost Hermitian structure;Einstein-Hilbert functional

#### References

1. D. E. Blair and S. Ianus, Critical associated metrics on symplectic manifolds, Nonlinear problems in geometry (Mobile, Ala., 1985), 23–29, Contemp. Math., 51, Amer. Math. Soc., Providence, RI, 1986. https://doi.org/10.1090/conm/051/848929
2. A. Gray, The structure of nearly K¨ahler manifolds, Math. Ann. 223 (1976), no. 3, 233–248. https://doi.org/10.1007/BF01360955
3. A. Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tohoku Math. J. (2) 28 (1976), no. 4, 601–612. https://doi.org/10.2748/tmj/1178240746
4. A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35–58. https://doi.org/10.1007/BF01796539
5. D. Hilbert, Die Grundlagen der Physik, Nachr. Ges. Wiss. Gott. (1915), 395–407.
6. T. Koda, Critical almost Hermitian structures, Indian J. Pure Appl. Math. 26 (1995), no. 7, 679–690.
7. S. Koto, Some theorems on almost Kahlerian spaces, J. Math. Soc. Japan 12 (1960), 422–433. https://doi.org/10.2969/jmsj/01240422
8. T. Oguro and K. Sekigawa, Some critical almost K¨ahler structures, Colloq. Math. 111 (2008), no. 2, 205–212. https://doi.org/10.4064/cm111-2-4
9. T. Oguro, K. Sekigawa, and A. Yamada, Some critical almost K¨ahler structures with a fixed Kahler class, Topics in contemporary differential geometry, complex analysis and mathematical physics, 269–277, World Sci. Publ., Hackensack, NJ, 2007.
10. K. Sekigawa, On some 4-dimensional compact almost Hermitian manifolds, J. Ramanujan Math. Soc. 2 (1987), no. 2, 101–116.
11. K. Sekigawa, Almost Hermitian manifolds satisfying some curvature conditions, Kodai Math. J. 2 (1979), no. 3, 384–405. https://doi.org/10.2996/kmj/1138036068

#### Cited by

1. CRITICAL HERMITIAN STRUCTURES ON THE PRODUCT OF SASAKIAN MANIFOLDS vol.09, pp.07, 2012, https://doi.org/10.1142/S0219887812500557
2. Some Critical Almost Hermitian Structures vol.63, pp.1-2, 2013, https://doi.org/10.1007/s00025-011-0143-8
3. Curvature identities derived from the integral formula for the first Pontrjagin number vol.31, pp.4, 2013, https://doi.org/10.1016/j.difgeo.2013.04.005
4. CURVATURE IDENTITIES DERIVED FROM AN INTEGRAL FORMULA FOR THE FIRST CHERN NUMBER vol.50, pp.4, 2013, https://doi.org/10.4134/BKMS.2013.50.4.1261