Metric of special \(2F\)-flat Riemannian spaces. (English) Zbl 1089.53020

The notion of \(F\)-planar mapping was introduced by J. Mikeš and N. S. Sinyukov [Izv. Vyssh. Uchebn. Zaved., Mat. 1983, No. 1(248), 55–61 (1983; Zbl 0514.53008)]. In this paper the author considers a \(2F\)-flat Riemannian space \(V_n\), that is a Riemannian space endowed with an affinor structure \(F\) such that \(F^3= I\), which is \(2F\)-planar mapped on flat space. He determines explicitly the metric tensor \(g\) of that Riemannian space \(V_n\).


53B20 Local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B35 Local differential geometry of Hermitian and Kählerian structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)


Zbl 0514.53008
Full Text: EuDML


[1] Beklemishev D. V.: Differential geometry of spaces with almost complex structure. Geometria. Itogi Nauki i Tekhn., All-Union Inst. for Sci. and Techn. Information (VINITI), Akad. Nauk SSSR, Moscow, 1965, 165-212.
[2] Eisenhart L. P.: Riemannian Geometry. : Princenton Univ. Press. 1926. · Zbl 0174.53303
[3] Kurbatova I. N.: HP-mappings of H-spaces. Ukr. Geom. Sb., Kharkov, 27 (1984), 75-82. · Zbl 0571.58006
[4] Al Lamy R. J.: About 2F-plane mappings of affine connection spaces. Coll. on Diff. Geom., Eger (Hungary), 1989, 20-25.
[5] Al Lamy R. J., Kurbatova I. N.: Invariant geometric objects of 2F-planar mappings of affine connection spaces and Riemannian spaces with affine structure of III order. Dep. of UkrNIINTI (Kiev), 1990, No. 1004Uk90, 51p.
[6] Al Lamy R. J., Mikeš J., Škodová M.: On linearly \(pF\)-planar mappings. Diff. Geom. and its Appl. Proc. Conf. Prague, 2004, Charles Univ., Prague (Czech Rep.), 2005, 347-353. · Zbl 1114.53012
[7] J. Mikeš: On special F-planar mappings of affine-connected spaces. Vestn. Mosk. Univ., 1994, 3, 18-24.
[8] Mikeš J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci., New York, 78, 3 (1996), 311-333. · Zbl 0866.53028
[9] Mikeš J.: Holomorphically projective mappings and their generalizations. J. Math. Sci., New York, 89, 3 (1998), 1334-1353. · Zbl 0983.53013
[10] Mikeš J., Pokorná O.: On holomorphically projective mappings onto Kählerian spaces. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 69, (2002), 181-186. · Zbl 1023.53015
[11] Mikeš J., Sinyukov N. S.: On quasiplanar mappings of spaces of affine connection. Sov. Math. 27, 1 (1983), 63-70; · Zbl 0526.53013
[12] Petrov A. Z.: New Method in General Relativity Theory. : Nauka, Moscow. 1966.
[13] Petrov A. Z.: Simulation of physical fields. Gravitation and the Theory of Relativity, Vol. 4-5, Kazan’ State Univ., Kazan, 1968, 7-21.
[14] Shirokov P. A.: Selected Work in Geometry. : Kazan State Univ. Press, Kazan. 1966.
[15] Sinyukov N. S.: Geodesic Mappings of Riemannian Spaces. : Nauka, Moscow. 1979. · Zbl 0637.53020
[16] Sinyukov N. S.: Almost geodesic mappings of affinely connected and Riemannian spaces. J. Sov. Math. 25 (1984), 1235-1249. · Zbl 0533.53014
[17] Škodová M., Mikeš J., Pokorná O.: On holomorphically projective mappings from equiaffine symmetric and recurrent spaces onto Kählerian spaces. Circ. Mat. di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 75, (2005), 309-316. · Zbl 1109.53019
[18] Yano K.: Differential Geometry on Complex, Almost Complex Spaces. : Pergamon Press, Oxford-London-New York-Paris-Frankfurt. XII, 1965, 323p. · Zbl 0127.12405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.