# Article

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Keywords:
$F^\varepsilon _2$-planar mapping; $PQ^\varepsilon$-projective equivalence; $F$-planar mapping; fundamental equation; (pseudo-) Riemannian manifold
Summary:
We study special $F$-planar mappings between two $n$-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced $PQ^{\varepsilon }$-projectivity of Riemannian metrics, $\varepsilon \ne 1,1+n$. Later these mappings were studied by Matveev and Rosemann. They found that for $\varepsilon =0$ they are projective. We show that $PQ^{\varepsilon }$-projective equivalence corresponds to a special case of $F$-planar mapping studied by Mikeš and Sinyukov (1983) and ${F_2}$-planar mappings (Mikeš, 1994), with $F=Q$. Moreover, the tensor $P$ is derived from the tensor $Q$ and the non-zero number $\varepsilon$. For this reason we suggest to rename $PQ^{\varepsilon }$ as ${F_2^{\varepsilon }}$. We use earlier results derived for ${F}$- and ${F_2}$-planar mappings and find new results. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.
References:
[1] Chudá, H., Shiha, M.: Conformal holomorphically projective mappings satisfying a certain initial condition. Miskolc Math. Notes 14 (2) (2013), 569–574. MR 3144093 | Zbl 1299.53037
[2] Hinterleitner, I.: On holomorphically projective mappings of e-Kähler manifolds. Arch. Mat. (Brno) 48 (2012), 333–338. DOI 10.5817/AM2012-5-333 | MR 3007616 | Zbl 1289.53038
[3] Hinterleitner, I., Mikeš, J.: On $F$-planar mappings of spaces with affine connections. Note Mat. 27 (2007), 111–118. MR 2367758
[4] Hinterleitner, I., Mikeš, J.: Fundamental equations of geodesic mappings and their generalizations. J. Math. Sci. 174 (5) (2011), 537–554. DOI 10.1007/s10958-011-0316-8
[5] Hinterleitner, I., Mikeš, J.: Projective equivalence and spaces with equi-affine connection. J. Math. Sci. 177 (2011), 546–550, transl. from Fundam. Prikl. Mat. 16 (2010), 47–54. DOI 10.1007/s10958-011-0479-3 | MR 2786490
[6] Hinterleitner, I., Mikeš, J.: Geodesic Mappings and Einstein Spaces. Geometric Methods in Physics, Birkhäuser Basel, 2013, arXiv: 1201.2827v1 [math.DG], 2012. MR 3364052 | Zbl 1268.53049
[7] Hinterleitner, I., Mikeš, J.: On holomorphically projective mappings from manifolds with equiaffine connection onto Kähler manifolds. Arch. Math. (Brno) 49 (5) (2013), 295–302. DOI 10.5817/AM2013-5-295 | MR 3159328
[8] Hinterleitner, I., Mikeš, J., Stránská, J.: Infinitesimal $F$-planar transformations. Russ. Math. 52 (2008), 13–18, transl. from Izv. Vyssh. Uchebn. Zaved., Mat. (2008), 16–22. DOI 10.3103/S1066369X08040026 | MR 2445169
[9] Hrdina, J.: Almost complex projective structures and their morphisms. Arch. Mat. (Brno) 45 (2009), 255–264. MR 2591680 | Zbl 1212.53022
[10] Hrdina, J., Slovák, J.: Generalized planar curves and quaternionic geometry. Ann. Global Anal. Geom. 29 (4) (2006), 349–360. DOI 10.1007/s10455-006-9023-y | MR 2251428
[11] Hrdina, J., Slovák, J.: Morphisms of almost product projective geometries. Proc. 10th Int. Conf. on Diff. Geom. and its Appl., DGA 2007, Olomouc. Hackensack, NJ: World Sci., 2008, pp. 253–261. MR 2462798
[12] Hrdina, J., Vašík, P.: Generalized geodesics on almost Cliffordian geometries. Balkan J. Geom. Appl. 17 (1) (2012), 41–48. MR 2911954 | Zbl 1284.53031
[13] Jukl, M., Juklová, L., Mikeš, J.: Some results on traceless decomposition of tensors. J. Math. Sci. (New York) 174 (2011), 627–640. DOI 10.1007/s10958-011-0321-y
[14] Lami, R.J.K. al, Škodová, M., Mikeš, J.: On holomorphically projective mappings from equiaffine generally recurrent spaces onto Kählerian spaces. Arch. Math. (Brno) 42 (5) (2006), 291–299. MR 2322415 | Zbl 1164.53317
[15] Levi-Civita, T.: Sulle transformationi delle equazioni dinamiche. Ann. Mat. Milano 24 Ser. 2 (1886), 255–300.
[16] Matveev, V., Rosemann, S.: Two remarks on $PQ^{\varepsilon }$-projectivity of Riemanninan metrics. Glasgow Math. J. 55 (1) (2013), 131–138. DOI 10.1017/S0017089512000390 | MR 3001335
[17] Mikeš, J.: On holomorphically projective mappings of Kählerian spaces. Ukr. Geom. Sb., Kharkov 23 (1980), 90–98. Zbl 0463.53013
[18] Mikeš, J.: Special $F$-planar mappings of affinely connected spaces onto Riemannian spaces. Mosc. Univ. Math. Bull. 49 (1994), 15–21, translation from Vestn. Mosk. Univ., Ser. 1 (1994), 18–24. MR 1315721 | Zbl 0896.53035
[19] Mikeš, J.: Holomorphically projective mappings and their generalizations. J. Math. Sci. (New York) 89 (1998), 1334–1353. DOI 10.1007/BF02414875 | MR 1619720
[20] Mikeš, J., Chudá, H., Hinterleitner, I.: Conformal holomorphically projective mappings of almost Hermitian manifolds with a certain initial condition. Int. J. Geom. Methods in Modern Phys. 11 (5) (2014), Article Number 1450044. DOI 10.1142/S0219887814500443 | MR 3208853
[21] Mikeš, J., Pokorná, O.: On holomorphically projective mappings onto almost Hermitian spaces. 8th Int. Conf. Opava, 2001, pp. 43–48. MR 1978761 | Zbl 1076.53506
[22] Mikeš, J., Pokorná, O.: On holomorphically projective mappings onto Kählerian spaces. Rend. Circ. Mat. Palermo (2) Suppl. 69 (2002), 181–186. MR 1972433
[23] Mikeš, J., Shiha, M., Vanžurová, A.: Invariant objects by holomorphically projective mappings of Kähler space. 8th Int. Conf. APLIMAT 2009: 8th Int. Conf. Proc., 2009, pp. 439–444.
[24] Mikeš, J., Sinyukov, N.S.: On quasiplanar mappings of space of affine connection. Sov. Math. (1983), 63–70, translation from Izv. Vyssh. Uchebn. Zaved., Mat. (1983), 55–61. MR 0694014
[25] Mikeš, J., Vanžurová, A., Hinterleitner, I.: Geodesic Mappings and some Generalizations. Palacky University Press, Olomouc, 2009. MR 2682926
[26] Otsuki, T., Tashiro, Y.: On curves in Kaehlerian spaces. Math. J. Okayama Univ. 4 (1954), 57–78. MR 0066024 | Zbl 0057.14101
[27] Petrov, A.Z.: Simulation of physical fields. Gravitatsiya i Teor. Otnositenosti 4–5 (1968), 7–21. MR 0285249
[28] Prvanović, M.: Holomorphically projective transformations in a locally product space. Math. Balkanica (N.S.) 1 (1971), 195–213. MR 0288710
[29] Sinyukov, N.S.: Geodesic Mappings of Riemannian Spaces. Moscow: Nauka, 1979, 256pp. MR 0552022 | Zbl 0637.53020
[30] Škodová, M., Mikeš, J., Pokorná, O.: On holomorphically projective mappings from equiaffine symmetric and recurrent spaces onto Kählerian spaces. Rend. Circ. Mat. Palermo (2) Suppl., vol. 75, 2005, pp. 309–316. MR 2152369 | Zbl 1109.53019
[31] Stanković, M.S., Zlatanović, M.L., Velimirović, L.S.: Equitorsion holomorphically projective mappings of generalized Kählerian space of the first kind. Czechoslovak Math. J. 60 (2010), 635–653. DOI 10.1007/s10587-010-0059-6 | MR 2672406 | Zbl 1224.53031
[32] Topalov, P.: Geodesic compatibility and integrability of geodesic flows. J. Math. Phys. 44 (2) (2003), 913–929. DOI 10.1063/1.1526939 | MR 1953103 | Zbl 1061.37042
[33] Yano, K.: Differential geometry on complex and almost complex spaces. vol. XII, Pergamon Press, Oxford-London-New York-Paris-Frankfurt, 1965, 323pp. MR 0187181 | Zbl 0127.12405

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