On Bochner flat para-Kählerian manifolds

• Autorzy: Dorota Łuczyszyn
• Języki publikacji: EN

Abstrakty

EN

Let B be the Bochner curvature tensor of a para-Kählerian manifold. It is proved that if the manifold is Bochner parallel (∇ B = 0), then it is Bochner flat (B = 0) or locally symmetric (∇ R = 0). Moreover, we define the notion of tha paraholomorphic pseudosymmetry of a para-Kählerian manifold. We find necessary and sufficient conditions for a Bochner flat para-Kählerian manifold to be paraholomorphically pseudosymmetric. Especially, in the case when the Ricci operator is diagonalizable, a Bochner flat para-Kählerian manifold is paraholomorphically pseudosymmetric if and only if the Ricci operator has at most two eigenvalues. A class of examples of manifolds of this kind is presented.

Słowa kluczowe

EN

Para-Kählerian manifold; Bochner conformal curvature; Bochner flat manifold; 53C15; 53C50; 53C56

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 2
• Strony: 331-341

Daty

wydano

2005-06-01

online

2005-06-01

Twórcy

autor

Dorota Łuczyszyn

• Wrocław University of Technology

Bibliografia

• [1] C.L. Bejan: “The Bochner curvature tensor on a hyperbolic Kähler manifold”, In: Colloquia Mathematica Societatis Jànos Bolyai, Vol. 56, Differential Geometry, Eger (Hungary), 1989, pp. 93–99,
• [2] A. Bonome, R. Castro, E. García-Río, L. Hervella and R. Vázquez-Lorenzo: “On the paraholomorphic sectional curvature of almost para-Hermitian manifolds”, Houston J. Math., Vol. 24, (1998), pp. 277–300.
• [3] R.L. Bryant: “Bochner-Kähler metrics”, J. Amer. Math. Soc., Vol. 14(3), (2001), pp. 623–715. http://dx.doi.org/10.1090/S0894-0347-01-00366-6
• [4] V. Cruceanu, P. Fortuny and P.M. Gadea: “A survey on paracomplex geometry”, Rocky Mountain J. Math., Vol. 26, (1996), pp. 83–115. http://dx.doi.org/10.1216/rmjm/1181072105
• [5] P.M. Gadea, V. Cruceanu and J. Muñoz Masqué: “Para-Hermitian and para-Kähler manifolds”, Quaderni Inst. Mat., Fac. Economia, Univ. Messina, Vol. 1, (1995), pp. 72.
• [6] G. Ganchev and A. Borisov: “Isotropic sections and curvature properties of hyperbolic Kaehlerian manifolds”, Publ. Inst. Math., Vol. 38, (1985), pp. 183–192.
• [7] E. García-Río, L. Hervella and R. Vázquez-Lorenzo: “Curvature properties of para-Kähler manifolds”, In: New developments in differential geometry (Debrecen, 1994), Math. Appl., Vol. 350, Kluwer Acad. Publ., Dordrecht, 1996, pp. 193–200.
• [8] S. Kaneyuki and M. Kozai: “Paracomplex structures and affine symmetric spaces”, Tokyo Math. J., Vol. 8, (1985), pp. 81–98. http://dx.doi.org/10.3836/tjm/1270151571
• [9] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, Vol. I, II, John Wiley & Sons, New York-London, 1963, 1969.
• [10] D. Luczyszyn: “On Bochner semisymmetric para-Kählerian manifolds”, Demonstr. Math., Vol. 34, (2001), pp. 933–942.
• [11] D. Łuczyszyn: “On pseudosymmetric para-Kählerian manifolds”, Beiträge Alg. Geom., Vol. 44, (2003), pp. 551–558.
• [12] M. Matsumoto and S. Tanno: “Kählerian spaces with parallel or vanishing Bochner curvature tensor”, Tensor N.S., Vol. 27, (1973), pp. 291–294.
• [13] Z. Olszak: “Bochner flat Kählerian manifolds”, In: Differential Geometry, Banach Center Publication, Vol. 12, PWN-Polish Scientific Publishers, Warsaw, 1984, pp. 219–223.
• [14] Z. Olszak: “Bochner flat Kählerian manifolds with a certain condition on the Ricci tensor”, Simon Stevin, Vol. 63, (1989), pp. 295–303.
• [15] E.M. Patterson: “Riemann extensions which have Kähler metrics”, Proc. Roy. Soc. Edinburgh (A), Vol. 64, (1954), pp. 113–126.
• [16] E.M. Patterson: “Symmetric Kähler spaces”, J. London Math. Soc., Vol. 30, (1955), pp. 286–291.
• [17] N. Pušić: “On an invariant tensor of a conformal transformation of a hyperbolic Kaehlerian manifold”, Zb. Rad. Fil. Fak. Niš, Ser. Mat., Vol. 4, (1990), pp. 55–64.
• [18] N. Pušić: “On HB-parallel hyperbolic Kaehlerian spaces”, Math. Balkanica N.S., Vol. 8, (1994), pp. 131–150.
• [19] N. Pušić: “On HB-recurrent hyperbolic Kaehlerian spaces”, Publ. Inst. Math. (Beograd) N.S., Vol. 55, (1994), pp. 64–74.
• [20] N. Pušić: “On HB-flat hyperbolic Kaehlerian spaces”, Mat. Vesnik, Vol. 49, (1997), pp. 35–44.
• [21] R.O. Wells: Differential analysis on complex manifolds, Graduate Texts in Mathematics, Vol. 65, Springer-Verlag, New York-Berlin, 1980.

Identyfikatory

DOI

10.2478/BF02499218