# On the bochner conformal curvature of Kähler-Norden manifolds

Open Mathematics (2005)

- Volume: 3, Issue: 2, page 309-317
- ISSN: 2391-5455

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topKarina Olszak. "On the bochner conformal curvature of Kähler-Norden manifolds." Open Mathematics 3.2 (2005): 309-317. <http://eudml.org/doc/268784>.

@article{KarinaOlszak2005,

abstract = {Using the one-to-one correspondence between Kähler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kähler-Norden manifold to be holomorphically recurrent.},

author = {Karina Olszak},

journal = {Open Mathematics},

keywords = {53C15; 53C50; 53C56},

language = {eng},

number = {2},

pages = {309-317},

title = {On the bochner conformal curvature of Kähler-Norden manifolds},

url = {http://eudml.org/doc/268784},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Karina Olszak

TI - On the bochner conformal curvature of Kähler-Norden manifolds

JO - Open Mathematics

PY - 2005

VL - 3

IS - 2

SP - 309

EP - 317

AB - Using the one-to-one correspondence between Kähler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kähler-Norden manifold to be holomorphically recurrent.

LA - eng

KW - 53C15; 53C50; 53C56

UR - http://eudml.org/doc/268784

ER -

## References

top- [1] A. Borowiec, M. Ferraris, M. Francaviglia and I. Volovich: “Almost-complex and almost-product Einstein manifolds from a variational principle”, J. Math. Physics, Vol. 40(7), (1999), pp. 3446–3464. http://dx.doi.org/10.1063/1.532899 Zbl0977.53042
- [2] A. Borowiec, M. Francaviglia and I. Volovich: “Anti-Kählerian manifolds”, Diff. Geom. Appl., Vol. 12, (2000), pp. 281–289. http://dx.doi.org/10.1016/S0926-2245(00)00017-6 Zbl0972.53043
- [3] E.J. Flaherty, Jr.: “The nonlinear gravitation in interaction with a photon”, General Relativity and Gravitation, Vol. 9(11), (1978), pp. 961–978. http://dx.doi.org/10.1007/BF00784657 Zbl0411.53014
- [4] A. Derdziński: “On homogeneous conformally symmetric pseudo-Riemannian manifolds”, Colloq. Math., Vol. 40, (1978), pp. 167–185. Zbl0418.53014
- [5] A. Derdziński: “The local structure of essentially conformally symmetric manifolds with constant fundamental function, I. The elliptic case, II. The hyperbolic case, III. The parabolic case”, Colloq. Math., Vol. 42, (1979), pp. 59–81; Vol. 44, (1981), pp. 77–95; Vol. 44, (1981), pp. 249–262. Zbl0439.53034
- [6] G.T. Ganchev and A.V. Borisov: “Note on the almost complex manifolds with Norden metric”, Compt. Rend. Acad. Bulg. Sci., Vol. 39, (1986), pp. 31–34. Zbl0608.53031
- [7] G.T. Ganchev, K. Gribachev and V. Mihova: “B-connections and their conformal invariants on conformally Kaehler manifolds with B-metric”, Publ. Inst. Math., Vol. 42(56), (1987), pp. 107–121. Zbl0638.53021
- [8] G. Ganchev and S. Ivanov: “Connections and curvatures on complex Riemannian manifolds”, Internal Report, No. IC/91/41, International Centre for Theoretical Physics, Trieste, Italy, 1991. Zbl0795.53065
- [9] G.T. Ganchev and S. Ivanov: “Characteristic curvatures on complex Riemannian manifolds”, Riv. Math. Univ. Parma (5), Vol. 1, (1992), pp. 155–162. Zbl0795.53065
- [10] S. Ivanov: “Holomorphically projective transformations on complex Riemannian manifold”, J. Geom., Vol. 49, (1994), pp. 106–116. http://dx.doi.org/10.1007/BF01228055 Zbl0823.53052
- [11] S. Kobayashi and K. Nomizu: Foundations of differential geometry, Vol. I, II, Interscience Publishers, New York, 1963, 1969. Zbl0119.37502
- [12] C.R. LeBrun: “H-space with a cosmological constant”, Proc. Roy. Soc. London, Ser. A, Vol. 380, (1982), pp. 171–185. http://dx.doi.org/10.1098/rspa.1982.0035
- [13] C. LeBrun: “Spaces of complex null geodesics in complex-Riemannian geometry”, Trans. Amer. Math. Soc., Vol. 278, (1983), pp. 209–231. http://dx.doi.org/10.2307/1999312 Zbl0562.53018
- [14] Z. Olszak: “On conformally recurrent manifolds, II. Riemann extensions”, Tensor N.S., Vol. 49, (1990), pp. 24–31. Zbl0841.53034
- [15] W. Roter: “On a class of conformally recurrent manifolds”, Tensor N.S., Vol. 39, (1982), pp. 207–217. Zbl0518.53018
- [16] W. Roter: “On the existence of certain conformally recurrent metrics”, Colloq. Math., Vol. 51, (1987), pp. 315–327. Zbl0622.53012
- [17] K. Sluka: “On Kähler manifolds with Norden metrics”, An. Stiint. Univ. “Al.I. Cuza” Ia§i, Ser. Ia Mat., Vol. 47 (2001), pp. 105–122.
- [18] K. Sluka: “Properties of the Weyl conformal curvature of Kähler-Norden manifolds”, In: Steps in Differental Geometry (Proc. Colloq. Diff. Geom. July 25–30, 2000), Debrecen, 2001, pp. 317–328.
- [19] K. Sluka: “On the curvature of Kähler-Norden manifolds”, J. Geom. Physics, (2004), in print.
- [20] Y.C. Wong: “Linear connexions with zero torsion and recurrent curvature”, Trans. Amer. Math. Soc., Vol. 102, (1962), pp. 471–506. http://dx.doi.org/10.2307/1993618 Zbl0139.39702
- [21] N. Woodhouse: “The real geometry of complex space-times”, Int. J. Theor. Phys., Vol. 16, (1977), pp. 663–670. http://dx.doi.org/10.1007/BF01812224 Zbl0385.53035

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