Hermitian *-Einstein surfaces

Grantcharov, Geo; Muškarov, Oleg

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

Hermitian $*$-Einstein surfaces
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by Geo Grantcharov and Oleg Muškarov PDF

Proc. Amer. Math. Soc. 120 (1994), 233-239 Request permission

Abstract:

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

References

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, DOI 10.1007/978-3-540-74311-8
Andrzej Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405–433. MR 707181
S. I. Goldberg, Integrability of almost Kaehler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100. MR 238238, DOI 10.1090/S0002-9939-1969-0238238-1
Alfred Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tohoku Math. J. (2) 28 (1976), no. 4, 601–612. MR 436054, DOI 10.2748/tmj/1178240746
Takashi Koda, Self-dual and anti-self-dual Hermitian surfaces, Kodai Math. J. 10 (1987), no. 3, 335–342. MR 929993, DOI 10.2996/kmj/1138037464
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
Zbigniew Olszak, On the existence of generalized complex space forms, Israel J. Math. 65 (1989), no. 2, 214–218. MR 998671, DOI 10.1007/BF02764861
Kouei Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 39 (1987), no. 4, 677–684. MR 905633, DOI 10.2969/jmsj/03940677
Kouei Sekigawa, On some $4$-dimensional compact almost Hermitian manifolds, J. Ramanujan Math. Soc. 2 (1987), no. 2, 101–116. MR 945613
K. Sekigawa and L. Vanhecke, Four-dimensional almost Kähler Einstein manifolds, Ann. Mat. Pura Appl. (4) 157 (1990), 149–160. MR 1108474, DOI 10.1007/BF01765316
Franco Tricerri and Lieven Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), no. 2, 365–397. MR 626479, DOI 10.1090/S0002-9947-1981-0626479-0
Izu Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc. 262 (1980), no. 2, 533–542. MR 586733, DOI 10.1090/S0002-9947-1980-0586733-7