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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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)
Acknowledgements
The authors would like to express their gratitude to the referee for valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
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