Abstract
We show that if a torsion tensor of anti-Hermitian metric connection is pure, then the given anti-Hermitian manifold is anti-Kähler. We prove that if an anti-Hermitian manifold is a conformally flat anti-Kähler–Codazzi manifold, then the scalar curvature vanishes, if and only if the given manifold is isotropic anti-Kähler. We also consider anti-Hermitian metrics of Hessian type defined by holomorphic Hamiltonian functions. Finally, we consider an example of anti-Kähler metrics on Walker 4-manifold.
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References
Bejan, C.-L., Druta-Romaniuc, S.-L.: Harmonic functions and quadratic harmonic morphisms on Walker space. Turk. J. Math. 40(5), 1004–1019 (2016)
Borowiec, A., Francaviglia, M., Volovich, I.: Anti-Kählerian manifolds. Diff. Geom. Appl. 12(3), 281–289 (2000)
Etayo, F., Santamaria, R.: (J2 \(=\pm \)1)-metric manifolds. Publ. Math. Debrecen. 57(3–4), 435–444 (2000)
Fernandez-Culma, E.A., Godoy, Y.: Anti-Kahlerian geometry on Lie groups. Math. Phys. Anal. Geom. 21(1), 8 (2018)
Ganchev, G.T., Borisov, A.V.: Note on the almost complex manifolds with a Norden metric. C. R. Acad. Bulgare Sci. 39(5), 31–34 (1986)
Iscan, M., Salimov, A.: On Kähler-Norden manifolds. Proc. Indian Acad. Sci. (Math. Sci.) 119(1), 71–80 (2009)
Manev, M.: Associated Nijenhuis tensors on manifolds with almost hypercomplex structures and metrics of Hermitian-Norden type. Results Math. 71(3–4), 1327–1343 (2017)
Matsushita, Y.: Four-dimensional Walker metrics and symplectic structures. J. Geom. Phys. 52(1), 89–99 (2004)
Matsushita, Y.: Walker 4-manifolds with proper almost complex structures. J. Geom. Phys. 55(4), 71–80 (2005)
Mekerov, D.: Connection with parallel totally skew-symmetric torsion on almost complex manifolds with Norden metric. C. R. Acad. Bulgare Sci. 62(12), 1501–1508 (2009)
Nannicini, A.: Generalized geometry of Norden manifolds. J. Geom. Phys. 99, 244–255 (2016)
Oproiu, V.: General natural almost Hermitian and anti-Hermitian structures on the tangent bundles. Bull. Math. Soc. Sci. Math. Roumanie 43(93), 325–340 (2000)
Salimov, A.: Almost psi-holomorphic tensors and their properties. Dokl. Acad. Nauk. 324(3), 533–536 (1992)
Salimov, A.: On operators associated with tensor fields. J. Geom. 99(1–2), 107–145 (2010)
Salimov, A., Turanli, S.: Curvature properties of anti-Kähler-Codazzi manifolds. C. R. Math. Acad. Sci. Paris 351(5–6), 225–227 (2013)
Salimov, A., Akbulut, K., Turanli, S.: On an isotropic property of anti-Kähler-Codazzi manifolds. C. R. Math. Acad. Sci. Paris 351(21–22), 837–839 (2013)
Salimov, A.: On anti-Hermitian metric connections. C. R. Math. Acad. Sci. Paris 352(9), 731–735 (2014)
Sluka, K.: On the curvature of Kahler-Norden manifolds. J. Geom. Phys. 54(2), 131–145 (2005)
Tachibana, S.: Analytic tensor and its generalization. Tohoku Math. J. 12(2), 208–221 (1960)
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The authors would like to express their gratitude to the referee for valuable suggestions and comments.
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Salimov, A., Azizova, S. Some Remarks Concerning Anti-Hermitian Metrics. Mediterr. J. Math. 16, 84 (2019). https://doi.org/10.1007/s00009-019-1358-2
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DOI: https://doi.org/10.1007/s00009-019-1358-2
Keywords
- Anti-Kähler–Codazzi manifold
- isotropic anti-Kähler manifold
- holomorphic function
- metric connection with torsion
- Walker metric