Hermitian *-Einstein surfaces

Grantcharov, Geo; Muškarov, Oleg

doi:10.1090/S0002-9939-1994-1186132-9

Hermitian -Einstein surfaces

Authors: Geo Grantcharov and Oleg Muškarov
Journal: Proc. Amer. Math. Soc. 120 (1994), 233-239
MSC: Primary 53C25; Secondary 53C55
DOI: https://doi.org/10.1090/S0002-9939-1994-1186132-9
MathSciNet review: 1186132
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the problem when a compact Hermitian ${\ast}$ -Einstein surface is Kählerian and show that it is true if is additionally assumed to be either Einstein or anti-self-dual. We also prove that if the ${\ast}$ -scalar curvature of is positive then is a conformally Kähler surface with positive first Chern class.

[B] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
[D] Andrzej Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405–433. MR 707181
[G0] S. I. Goldberg, Integrability of almost Kaehler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100. MR 238238, https://doi.org/10.1090/S0002-9939-1969-0238238-1
[Gr] Alfred Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tohoku Math. J. (2) 28 (1976), no. 4, 601–612. MR 436054, https://doi.org/10.2748/tmj/1178240746
[K] Takashi Koda, Self-dual and anti-self-dual Hermitian surfaces, Kodai Math. J. 10 (1987), no. 3, 335–342. MR 929993, https://doi.org/10.2996/kmj/1138037464
[Ol] Zbigniew Olszak, A note on almost Kaehler manifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 2, 139–141 (English, with Russian summary). MR 493830
[O2] Zbigniew Olszak, On the existence of generalized complex space forms, Israel J. Math. 65 (1989), no. 2, 214–218. MR 998671, https://doi.org/10.1007/BF02764861
[S1] Kouei Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 39 (1987), no. 4, 677–684. MR 905633, https://doi.org/10.2969/jmsj/03940677
[S2] Kouei Sekigawa, On some 4-dimensional compact almost Hermitian manifolds, J. Ramanujan Math. Soc. 2 (1987), no. 2, 101–116. MR 945613
[SV] K. Sekigawa and L. Vanhecke, Four-dimensional almost Kähler Einstein manifolds, Ann. Mat. Pura Appl. (4) 157 (1990), 149–160. MR 1108474, https://doi.org/10.1007/BF01765316
[TV] Franco Tricerri and Lieven Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), no. 2, 365–397. MR 626479, https://doi.org/10.1090/S0002-9947-1981-0626479-0
[V] Izu Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc. 262 (1980), no. 2, 533–542. MR 586733, https://doi.org/10.1090/S0002-9947-1980-0586733-7