Research Article
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Year 2016, , 37 - 46, 30.10.2016
https://doi.org/10.36753/mathenot.421451

Abstract

References

  • [1] 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 9(1994), 1457-1494. [2] Bielawski, R., Complexification and hypercomplexification of manifolds with a linear connection, Internat. J. Math. 14(2003), no.8, 813-824.
  • [3] Dancer, A. and Swann, A., Hypersymplectic manifolds, in "Recent developments in pseudo-Riemannian geometry". ESI Lectures in Mathematics and Physics (2008), 97-148.
  • [4] Davidov, J., Grantcharov, G., Mushkarov, O. and Yotov, M., Para-hyperhermitian surfaces. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(2009), no.3, 281-289.
  • [5] Dida, H.M., Hathout, F. and Djaa, M., On the geometry of the second order tangent bundle with the diagonal lift metric, Int. Journal of Math. Anal. 3(2009), 443-456.
  • [6] Djaa, M. and Gancarzewicz, J., The geometry of tangent bundles of order r, Boletin Academia Galega de Ciencias spain 4(1985), 147-165.
  • [7] Dodson, C.T.J. and Radivoiovici, M.S., Tangent and Frames Bundles of order two, Analele stiintifice ale Univesitatii ”AL. I Gusa” 28(1982), 63-71.
  • [8] Dombrowski, P., On the geometry of the tangent bundle. J.Reine Angew. Math. 210(1962), 73-88.
  • [9] Feix, B., Hyperkahler metrics on cotangent bundles, J. Reine Angew. Math 532(2001), 33-46.
  • [10] Hathout, F. , Dida, H.M., Diagonal lift in the tangent bundle of order two and its applications, Turk. J. Math. 29(2005), 1-12.
  • [11] Hull, C.M., Actions for (2,1) sigma models and strings, Nucl. Phys. B 509(1998), 252-272.
  • [12] Ivanov, S., Sanov, V.T. and Zamkovoy, S., Hyper-para-Hermitian manifolds with torsion, J. Geom. Phys. 56(2006), no.4, 670-690.
  • [13] Ivanov, S. and Zamkovoy, S., Para-hermitianand para-quaternionic manifolds. Differ. Geom. Appl. 23(2005), 205-234.
  • [14] Kaledin, D., A canonical hyperkähler metric on the total space of a cotangent bundle. Proceedings of the Second Meeting "Quaternionic structures in mathematics and physics" (Rome, 1999), World Sci. Publishing, River Edge, NJ (2001), 195-230.
  • [15] Kamada, H., Neutral hyper-Kähler structures on primary Kodaira surfaces, Tsukuba J. Math. 23(1999) 321-332.
  • [16] Lanus, S. and Vîlcu, G.E., Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles, Int. J. Geom. Methods Mod. Phys., 5(2008), no.6, 893-903.
  • [17] Lonescu, A.M. and Vîlcu, G.E., A Note on para-quaternionic Manifolds, Missouri J. Math. Sci., 19(2007), 213-218.
  • [18] Nakashima, Y. and Watanabe, Y., Some constructions of almost Hermitian and quaternion metric structures, Math. J. Toyama Univ., 13(1900), 119-138.
  • [19] Tahara, M., Marchiafava, S. and Watanabe, Y., Quaternionic Kähler structures on the tangent bundle of a complex space form, Rend. Ist. Mat. Univ. Trieste, 31(1999), 163-175.
  • [20] Vîlcu, G.E., Para-hyperhermitian structures on tangent bundles, Proceedings of the Estonian Academy of Sciences 60(2011), no.3, 165-173.

Para-Quaternionic Structures on the 3-Jet Bundle

Year 2016, , 37 - 46, 30.10.2016
https://doi.org/10.36753/mathenot.421451

Abstract

In this paper we construct an almost para-quaternionic structure on the 3-jet bundle of an almost parahermitian
manifold and we study its integrability. We give a necessary and sufficient conditions that are
provided for these structures to become para-hyper-Kähler and we prove that the 3-jet bundle can not be
a para-quaternionic Kähler manifold. 

References

  • [1] 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 9(1994), 1457-1494. [2] Bielawski, R., Complexification and hypercomplexification of manifolds with a linear connection, Internat. J. Math. 14(2003), no.8, 813-824.
  • [3] Dancer, A. and Swann, A., Hypersymplectic manifolds, in "Recent developments in pseudo-Riemannian geometry". ESI Lectures in Mathematics and Physics (2008), 97-148.
  • [4] Davidov, J., Grantcharov, G., Mushkarov, O. and Yotov, M., Para-hyperhermitian surfaces. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(2009), no.3, 281-289.
  • [5] Dida, H.M., Hathout, F. and Djaa, M., On the geometry of the second order tangent bundle with the diagonal lift metric, Int. Journal of Math. Anal. 3(2009), 443-456.
  • [6] Djaa, M. and Gancarzewicz, J., The geometry of tangent bundles of order r, Boletin Academia Galega de Ciencias spain 4(1985), 147-165.
  • [7] Dodson, C.T.J. and Radivoiovici, M.S., Tangent and Frames Bundles of order two, Analele stiintifice ale Univesitatii ”AL. I Gusa” 28(1982), 63-71.
  • [8] Dombrowski, P., On the geometry of the tangent bundle. J.Reine Angew. Math. 210(1962), 73-88.
  • [9] Feix, B., Hyperkahler metrics on cotangent bundles, J. Reine Angew. Math 532(2001), 33-46.
  • [10] Hathout, F. , Dida, H.M., Diagonal lift in the tangent bundle of order two and its applications, Turk. J. Math. 29(2005), 1-12.
  • [11] Hull, C.M., Actions for (2,1) sigma models and strings, Nucl. Phys. B 509(1998), 252-272.
  • [12] Ivanov, S., Sanov, V.T. and Zamkovoy, S., Hyper-para-Hermitian manifolds with torsion, J. Geom. Phys. 56(2006), no.4, 670-690.
  • [13] Ivanov, S. and Zamkovoy, S., Para-hermitianand para-quaternionic manifolds. Differ. Geom. Appl. 23(2005), 205-234.
  • [14] Kaledin, D., A canonical hyperkähler metric on the total space of a cotangent bundle. Proceedings of the Second Meeting "Quaternionic structures in mathematics and physics" (Rome, 1999), World Sci. Publishing, River Edge, NJ (2001), 195-230.
  • [15] Kamada, H., Neutral hyper-Kähler structures on primary Kodaira surfaces, Tsukuba J. Math. 23(1999) 321-332.
  • [16] Lanus, S. and Vîlcu, G.E., Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles, Int. J. Geom. Methods Mod. Phys., 5(2008), no.6, 893-903.
  • [17] Lonescu, A.M. and Vîlcu, G.E., A Note on para-quaternionic Manifolds, Missouri J. Math. Sci., 19(2007), 213-218.
  • [18] Nakashima, Y. and Watanabe, Y., Some constructions of almost Hermitian and quaternion metric structures, Math. J. Toyama Univ., 13(1900), 119-138.
  • [19] Tahara, M., Marchiafava, S. and Watanabe, Y., Quaternionic Kähler structures on the tangent bundle of a complex space form, Rend. Ist. Mat. Univ. Trieste, 31(1999), 163-175.
  • [20] Vîlcu, G.E., Para-hyperhermitian structures on tangent bundles, Proceedings of the Estonian Academy of Sciences 60(2011), no.3, 165-173.
There are 19 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Bouazza Kacimi This is me

Fouzi Hathout This is me

H.mohamed Dida This is me

Mokhtaria Barnoussi This is me

Publication Date October 30, 2016
Submission Date August 30, 2016
Published in Issue Year 2016

Cite

APA Kacimi, B., Hathout, F., Dida, H., Barnoussi, M. (2016). Para-Quaternionic Structures on the 3-Jet Bundle. Mathematical Sciences and Applications E-Notes, 4(2), 37-46. https://doi.org/10.36753/mathenot.421451
AMA Kacimi B, Hathout F, Dida H, Barnoussi M. Para-Quaternionic Structures on the 3-Jet Bundle. Math. Sci. Appl. E-Notes. October 2016;4(2):37-46. doi:10.36753/mathenot.421451
Chicago Kacimi, Bouazza, Fouzi Hathout, H.mohamed Dida, and Mokhtaria Barnoussi. “Para-Quaternionic Structures on the 3-Jet Bundle”. Mathematical Sciences and Applications E-Notes 4, no. 2 (October 2016): 37-46. https://doi.org/10.36753/mathenot.421451.
EndNote Kacimi B, Hathout F, Dida H, Barnoussi M (October 1, 2016) Para-Quaternionic Structures on the 3-Jet Bundle. Mathematical Sciences and Applications E-Notes 4 2 37–46.
IEEE B. Kacimi, F. Hathout, H. Dida, and M. Barnoussi, “Para-Quaternionic Structures on the 3-Jet Bundle”, Math. Sci. Appl. E-Notes, vol. 4, no. 2, pp. 37–46, 2016, doi: 10.36753/mathenot.421451.
ISNAD Kacimi, Bouazza et al. “Para-Quaternionic Structures on the 3-Jet Bundle”. Mathematical Sciences and Applications E-Notes 4/2 (October 2016), 37-46. https://doi.org/10.36753/mathenot.421451.
JAMA Kacimi B, Hathout F, Dida H, Barnoussi M. Para-Quaternionic Structures on the 3-Jet Bundle. Math. Sci. Appl. E-Notes. 2016;4:37–46.
MLA Kacimi, Bouazza et al. “Para-Quaternionic Structures on the 3-Jet Bundle”. Mathematical Sciences and Applications E-Notes, vol. 4, no. 2, 2016, pp. 37-46, doi:10.36753/mathenot.421451.
Vancouver Kacimi B, Hathout F, Dida H, Barnoussi M. Para-Quaternionic Structures on the 3-Jet Bundle. Math. Sci. Appl. E-Notes. 2016;4(2):37-46.

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