ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2020): 1.045

Para-hyperhermitian structures on tangent bundles; pp. 165–173

Full article in PDF format | doi: 10.3176/proc.2011.3.04

Author
Gabriel Eduard Vîlcu

Abstract
In this paper we construct a family of almost para-hyperhermitian structures on the tangent bundle of an almost para-hermitian manifold and study its integrability. Also, the necessary and sufficient conditions are provided for these structures to become para-hyper-Kähler.
References

  1. Andrada, A. and Dotti, G. I. Double products and hypersymplectic structures on R4n. Commun. Math. Phys., 2006, 262(1), 1–16.
doi:10.1007/s00220-005-1472-9

  2. Barret, J., Gibbons, G. W., Perry, M. J., Pope, C. N., and Ruback, P. Kleinian geometry and the N = 2 superstring. Int. J. Mod. Phys. A, 1994, 9, 1457–1494.
doi:10.1142/S0217751X94000650

  3. Bejan, C. L. and Oproiu, V. Tangent bundles of quasi-constant holomorphic sectional curvatures. Balkan J. Geom. Appl., 2006, 11(1), 11–22.

  4. Blair, D. E., Davidov, J., and Muškarov, O. Hyperbolic twistor space. Rocky Mt. J. Math., 2005, 35(5), 1437–1465.
doi:10.1216/rmjm/1181069645

  5. Cortés, V. The special geometry of Euclidian supersymmetry: a survey. Rev. Unión Mat. Argent., 2006, 47(1), 29–34.

  6. Cortés, V., Mayer, C., Mohaupt, T. and Saueressig, F. Special geometry of euclidean supersymmetry II. Hypermultiplets and the c-map. J. High Energy Phys., 2005, 6, 1–25.

  7. Dancer, A. and Swann, A. Hypersymplectic manifolds. In Recent Developments in Pseudo-Riemannian Geometry (Alekseevsky, D., ed.). ESI Lectures in Mathematics and Physics, 2008, 97–148.
doi:10.4171/051-1/3

  8. Davidov, J., Grantcharov, G., Mushkarov, O., and Yotov, M. Para-hyperhermitian surfaces. Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 2009, 52(100), 281–289.

  9. Dunajski, M. Hyper-complex four-manifolds from Tzitzeica equation. J. Math. Phys., 2002, 43, 651–658.
doi:10.1063/1.1426687

10. Dunajski, M. and West, S. Anti-self-dual conformal structures in neutral signature. In Recent Developments in Pseudo-Riemannian Geometry (Alekseevsky, D., ed.). ESI Lectures in Mathematics and Physics, 2008, 113–148.
doi:10.4171/051-1/4

11. Dombrowski, P. On the geometry of the tangent bundle. J. Reine Angew. Math., 1962, 210, 73–88.
doi:10.1515/crll.1962.210.73

12. Fino, A., Pedersen, H., Poon, Y.-S., and Sørensen, M. W. Neutral Calabi-Yau structures on Kodaira manifolds. Commun. Math. Phys., 2004, 248(2), 255–268.
doi:10.1007/s00220-004-1108-5

13. Hitchin, N. Hypersymplectic quotients. Acta Acad. Sci. Tauriensis, 1990, 124, 169–180.

14. Hull, C. M. Actions for (2,1) sigma models and strings. Nucl. Phys. B, 1998, 509, 252–272.
doi:10.1016/S0550-3213(97)00492-6

15. Ianuş, S. and Vîlcu, G. E. Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles. Int. J. Geom. Methods Mod. Phys., 2008, 5(6), 893–903.
doi:10.1142/S0219887808003016

16. Ii, K. and Morikawa, T. Kähler structures on the tangent bundle of Riemannian manifolds of constant positive curvature. Bull. Yamagata Univ. Nat. Sci., 1999, 14(3), 141–154.

17. Ivanov, S. and Zamkovoy, S. Para-hermitian and para-quaternionic manifolds. Differ. Geom. Appl., 2005, 23, 205–234.
doi:10.1016/j.difgeo.2005.06.002

18. Ivanov, S., Tsanov, V., and Zamkovoy, S. Hyper-para-Hermitian manifolds with torsion. J. Geom. Phys., 2006, 56(4), 670–690.
doi:10.1016/j.geomphys.2005.04.012

19. Kamada, H. Neutral hyper-Kähler structures on primary Kodaira surfaces. Tsukuba J. Math., 1999, 23, 321–332.

20. Libermann, P. Sur le problème d’équivalence de certaines structures infinitésimales. Ann. Mat. Purra Appl., 1954, 36, 27–120.
doi:10.1007/BF02412833

21. Nakashima, Y. and Watanabe, Y. Some constructions of almost Hermitian and quaternion metric structures. Math. J. Toyama Univ., 1990, 13, 119–138.

22. Olszak, Z. On almost complex structures with Norden metrics on tangent bundles. Period. Math. Hung., 2005, 51(2), 59–74.
doi:10.1007/s10998-005-0030-8

23. Ooguri, H. and Vafa, C. Geometry of N = 2 strings. Nucl. Phys. B, 1991, 361, 469–518.
doi:10.1016/0550-3213(91)90270-8

24. Oproiu, V. and Papaghiuc, N. General natural Einstein Kähler structures on tangent bundles. Differ. Geom. Appl., 2009, 27(3), 384–392.
doi:10.1016/j.difgeo.2008.10.017

25. Tahara, M., Vanhecke, L., and Watanabe, Y. New structures on tangent bundles. Note Mat., 1998, 18(1), 131–141.

26. Tahara, M., Marchiafava, S., and Watanabe, Y. Quaternionic Kähler structures on the tangent bundle of a complex space form. Rend. Ist. Mat. Univ. Trieste, 1999, 31(1–2), 163–175.
Back to Issue