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Keywords:
geodesic mapping; equitorsion geodesic mapping; generalized Kählerian space
geodesic mapping; equitorsion geodesic mapping; generalized Kählerian space
Summary:
In the present paper a generalized Kählerian space $\mathbb {G} {\underset 1 {\mathbb {K}}_N}$ of the first kind is considered as a generalized Riemannian space $\mathbb {GR}_N$ with almost complex structure $\smash {F^h_i}$ that is covariantly constant with respect to the first kind of covariant derivative. \endgraf Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings $f\colon \mathbb {GR}_N\to \mathbb {G}\underset 1{\mathbb {\overline {K}}}_N$ with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space $\mathbb {G}{\underset 1 {\mathbb {K}}}_N$.
References:
[1] Domašev, V. V., Mikeš, J.: On the theory of holomorphically projective mappings of Kählerian spaces. Math. Notes 23 (1978), 160-163 translated from Matematicheskie Zametki 23 (1978), Russian 297-303. DOI 10.1007/BF01153160 | MR 0492674
[2] Einstein, A.: The Meaning of Relativity. Princeton University Press Princeton, N. J. (1955). MR 0076496 | Zbl 0067.20404
[3] Einstein, A.: The Bianchi identities in the generalized theory of gravitation. Can. J. Math. 2 (1950), 120-128. DOI 10.4153/CJM-1950-011-4 | MR 0034134 | Zbl 0039.38802
[4] Einstein, A.: A generalization of the relativistic theory of gravitation. Ann. Math. (2) 46 (1945), 578-584. DOI 10.2307/1969197 | MR 0014296 | Zbl 0060.44113
[5] Eisenhart, L. P.: Generalized Riemann spaces. Proc. Natl. Acad. Sci. USA 37 (1951), 311-315. DOI 10.1073/pnas.37.5.311 | MR 0043530 | Zbl 0043.37301
[6] Hinterleitner, I., Mikeš, J.: On {$F$}-planar mappings of spaces with affine connections. Note Mat. 27 (2007), 111-118. MR 2367758 | Zbl 1150.53009
[7] Mikeš, J.: Holomorphically projective mappings and their generalizations. J. Math. Sci., New York 89 (1998), 1334-1353 translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory 30 (1995), Russian. DOI 10.1007/BF02414875 | MR 1619720
[8] Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci., New York 78 (1996), 311-333 translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory 11 (1994), Russian. DOI 10.1007/BF02365193 | MR 1384327
[9] Mikeš, J.: Geodesic mappings of Ricci 2-symmetric Riemannian spaces. Math. Notes 28 (1981), 922-924 translated from Matematicheskie Zametki 28 313-317 (1980), Russian. MR 0587405
[10] Mikeš, J., Starko, G. A.: $K$-concircular vector fields and holomorphically projective mappings on Kählerian spaces. Proceedings of the 16th Winter School on ``Geometry and Physics'', Srn'ı, Czech Republic, 1996 Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 46 Palermo (1997), 123-127 Jan Slovák et al. MR 1469028
[11] Mikeš, J., Vanžurová, A., Hinterleitner, I.: Geodesic Mappings and Some Generalizations. Palacký University, Faculty of Science Olomouc (2009). MR 2682926 | Zbl 1222.53002
[12] Minčić, S. M.: New commutation formulas in the non-symmetric affine connexion space. Publ. Inst. Math., Nouv. Sér. 22 (1977), 189-199. MR 0482552 | Zbl 0377.53008
[13] Minčić, S. M.: Ricci identities in the space of non-symmetric affine connexion. Mat. Vesn., N. Ser. 10 (1973), 161-172. MR 0341310 | Zbl 0278.53012
[14] Minčić, S. M., Stanković, M. S.: Equitorsion geodesic mappings of generalized Riemannian spaces. Publ. Inst. Math., Nouv. Sér. 61 (1997), 97-104. MR 1472941 | Zbl 0886.53035
[15] Minčić, S., Stanković, M.: On geodesic mappings of general affine connexion spaces and of generalized Riemannian spaces. Mat. Vesn. 49 (1997), 27-33. MR 1491944 | Zbl 0949.53013
[16] Minčić, S. M., Stanković, M. S., Velimirović, L. S.: Generalized Kählerian spaces. Filomat 15 (2001), 167-174. MR 2105108
[17] Moffat, J. W.: Gravitational theory, galaxy rotation curves and cosmology without dark matter. J. Cosmol. Astropart. Phys. (electronic only) 2005 (2005), Article No. 003, 28 pages. MR 2139872 | Zbl 1236.83045
[18] Ōtsuki, T., Tashiro, Y.: On curves in Kaehlerian spaces. Math. J. Okayama Univ. 4 (1954), 57-78. MR 0066024 | Zbl 0057.14101
[19] Prvanović, M.: A note on holomorphically projective transformations of the Kähler spaces. Tensor, New Ser. 35 (1981), 99-104. MR 0614141 | Zbl 0467.53032
[20] Pujar, S. S.: On non-metric semi-symmetric complex connection in a Kaehlerian manifold. Bull. Calcutta Math. Soc. 91 (1999), 313-322. MR 1748542 | Zbl 0957.53039
[21] Pušić, N.: On a curvature-type invariant of a family of metric holomorphically semi-symmetric connections on anti-Kähler spaces. Indian J. Math. 54 (2012), 57-74. MR 2976295 | Zbl 1268.53020
[22] Sinjukov, N. S.: Geodesic mappings of Riemannian spaces. Nauka Moskva Russian (1979). MR 0552022
[23] Stanković, M. S., Minčić, S. M., Velimirović, L. S.: On equitorsion holomorphically projective mappings of generalised Kählerian spaces. Czech. Math. J. 54 (2004), 701-715. DOI 10.1007/s10587-004-6419-3 | MR 2086727
[24] Stanković, M. S., Zlatanović, M. L., Velimirović, L. S.: Equitorsion holomorphically projective mappings of generalized Kählerian space of the first kind. Czech. Math. J. 60 (2010), 635-653. DOI 10.1007/s10587-010-0059-6 | MR 2672406 | Zbl 1224.53031
[25] Tashiro, Y.: On a holomorphically projective correspondence in an almost complex space. Math. J. Okayama Univ. 6 (1957), 147-152. MR 0087181 | Zbl 0077.35501
[26] Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. International Series of Monographs in Pure and Applied Mathematics 49 Pergamon Press, Macmillan, New York (1965). MR 0187181 | Zbl 0127.12405
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