Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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REMARKS ON LEVI HARMONICITY OF CONTACT SEMI-RIEMANNIAN MANIFOLDS

  • Perrone, Domenico (Universita del Salento Dipartimento di Matematica e Fisica "E. De Giorgi" Via Provinciale Lecce-Arnesano)
  • Received : 2012.12.18
  • Published : 2014.09.01
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Abstract

In a recent paper [10] we introduced the notion of Levi harmonic map f from an almost contact semi-Riemannian manifold (M, ${\varphi}$, ${\xi}$, ${\eta}$, g) into a semi-Riemannian manifold $M^{\prime}$. In particular, we compute the tension field ${\tau}_H(f)$ for a CR map f between two almost contact semi-Riemannian manifolds satisfying the so-called ${\varphi}$-condition, where $H=Ker({\eta})$ is the Levi distribution. In the present paper we show that the condition (A) of Rawnsley [17] is related to the ${\varphi}$-condition. Then, we compute the tension field ${\tau}_H(f)$ for a CR map between two arbitrary almost contact semi-Riemannian manifolds, and we study the concept of Levi pluriharmonicity. Moreover, we study the harmonicity on quasicosymplectic manifolds.

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References

  1. P. Baird and J. C.Wood, Harmonic morphisms between Riemannian manifolds, London Mathem. Society Monographs, Vol. 29, Oxford Science Publications, Clarendon Press, Oxford, 2003.
  2. E. Barletta, S. Dragomir, and H. Urakawa, Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds, Indiana Univ. Math. J. 50 (2001), no. 2, 719-746.
  3. E. Barros and A. Romero, Indefinite Kahler manifolds, Math. Ann. 261 (1982), no. 1, 55-62. https://doi.org/10.1007/BF01456410
  4. A. Bejancu and K. L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds, Int. J. Math. Math. Sci. 16 (1993), no. 3, 545-556. https://doi.org/10.1155/S0161171293000675
  5. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhauser, Boston, Basel, Berlin, 2002.
  6. G. Calvaruso and D. Perrone, Contact pseudo-metric manifolds, Differential Geom. Appl. 28 (2010), no. 5, 615-634. https://doi.org/10.1016/j.difgeo.2010.05.006
  7. M. Capursi, Quasicosymplectic manifolds, Rev. Roumaine Math. Pures Appl. 32 (1987), no. 1, 27-35.
  8. J. Davidov, Almost contact metric structures and twistor spaces, Houston J. Math. 29 (2003), no. 3, 639-673.
  9. S. Dragomir and D. Perrone, Harmonic Vector Fields: Variational Principles and Differential Geometry, Elsevier, Science Ltd, 2011.
  10. S. Dragomir and D. Perrone, Levi harmonic maps of contact Riemannian manifolds, J. Geome. Anal. 24 (2014), no. 3, 1233-1275. https://doi.org/10.1007/s12220-012-9371-8
  11. B. Fuglede, Harmonic morphisms between semi-Riemannian manifolds, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 31-50.
  12. Y. Ohnita, On pluriharmonicity of stable harmonic maps, J. London Math. Soc. (2) 35 (1987), no. 3, 563-568.
  13. Z. Olszak, On almost cosymplectic manifolds, Kodai Math. J. 4 (1981), no. 2, 239-250. https://doi.org/10.2996/kmj/1138036371
  14. D. Perrone, Classification of homogeneous almost cosymplectic three-manifolds, Differential Geom. Appl. 30 (2012), no. 1, 49-58. https://doi.org/10.1016/j.difgeo.2011.10.003
  15. D. Perrone, Curvature of K-contact semi-Riemannian manifolds, Canad. Math. Bull. 57 (2014), no. 2, 401-412. https://doi.org/10.4153/CMB-2013-016-7
  16. D. Perrone, Contact pseudo-metric manifolds of constant curvature and CR geometry, Results Math. 2014 Online first 2014, Springer Basel DOI 10.1007/s00025-014-0373-7.
  17. J. H. Rawnsley, f-structures, f-twistor spaces and harmonic maps, Geometry Seminar L. Bianchi, Lecture Notes in Math. 1164, 84-159, Springer-Verlag, 1985.
  18. K. Yano and S. Ishihara, Invariant submanifolds of an almost contact manifold, Kodai Math. Sem. Rep. 21 (1969), 350-364. https://doi.org/10.2996/kmj/1138845942
  19. K. Yano and M. Kon, Structures on Manifolds, World Scientific, Series in Pure Mathematics, vol. 3, 1984.

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