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

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

Gabriel Eduard Vîlcu

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.

  1. Andrada, A. and Dotti, G. I. Double products and hypersymplectic structures on R4n. Commun. Math. Phys., 2006, 262(1), 1–16.

  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.

  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.

  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.

  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.

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.

11. Dombrowski, P. On the geometry of the tangent bundle. J. Reine Angew. Math., 1962, 210, 73–88.

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.

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.

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.

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.

18. Ivanov, S., Tsanov, V., and Zamkovoy, S. Hyper-para-Hermitian manifolds with torsion. J. Geom. Phys., 2006, 56(4), 670–690.

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.

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.

23. Ooguri, H. and Vafa, C. Geometry of N = 2 strings. Nucl. Phys. B, 1991, 361, 469–518.

24. Oproiu, V. and Papaghiuc, N. General natural Einstein Kähler structures on tangent bundles. Differ. Geom. Appl., 2009, 27(3), 384–392.

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.
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