Abstract
In this paper, we present some results obtained for symmetric, semisymmetric, and semisymmetric recurrent projectively Euclidean spaces. Components of objects of affine connections of these spaces are found.
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V. E. Berezovskii, N. I. Guseva, and J. Mikeš, “On a particular case of first-type almost geodesic mappings of affinely connected spaces that preserve a certain tensor,” Mat. Zametki, 98, No. 3, 463–466 (2015).
V. Berezovskii, I. Hinterleitner, and J. Mikeš, “Geodesic mappings of manifolds with affine connection onto the Ricci symmetric manifolds,” Filomat, 32, No. 2, 379–385 (2018).
V. E. Berezovskii, L. E. Kovalev, and J. Mikeš, “On preserving of the Riemann tensor under some mappings of affinely connected spaces,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 9, 3-10 (2018).
É. Cartan, “Sur les variété à connexion projective,” Bull. Soc. Math. Fr., 52, 205–241 (1924).
É. Cartan, “Sur une classe remarquable d’espaces de Riemann, II,” Bull. Soc. Math. Fr., 55, 114–134 (1927).
L. P. Eisenhart, Non-Riemannian geometry, Princeton Univ. Press (1926).
I. Hinterleitner and J. Mikeš, “Geodesic mappings onto Weyl manifolds,” J. Appl. Math., 2, No. 1, 125–133 (2009).
I. Hinterleiner and J. Mikeš, “Fundamental equations of geodesic mappings and their generalizations,” J. Math. Sci., 174, 537–554 (2011).
I. Hinterleiner and J. Mikeš, “Projective equivalence and spaces with equiaffine connection,” J. Math. Sci., 177, 546–550 (2011).
I. Hinterleiner and J. Mikeš, “Geodesic mappings and Einstein spaces,” in: Geometric Methods in Physics, Birkhäuser, Basel (2013), pp. 331–335.
I. Hinterleiner and J. Mikeš, “Geodesic Mappings of (pseudo-) Riemannian manifolds preserve class of differentiability,” Miskolc. Math. Notes., 14, 89–96 (2013).
I. Hinterleiner and J. Mikeš, “Geodesic mappings and differentiability of metrics, affine and projective connections,” Filomat, 29, 1245–1249 (2015).
I. Hinterleiner and J.Mikeš, “Geodesic mappings onto Riemannian manifolds and differentiability,” in: Geometry, Integrability and Quantization. Vol. 18, Bulgar. Acad. Sci., Sofia (2017), pp. 183–190.
V. R. Kaigorodov, “Structure of the curvature of the space-time,” Itogi Nauki Tekhn. Ser. Probl. Geom., 14, 177-204 (1983).
P. I. Kovalev, “Triple Lie systems and spaces of affine connection,” Mat. Zametki, 14, No. 1, 107-112 (1973).
A. Lichnerowicz, “Courbure, nombres de Betti, et espaces symétriques,” in: Proc. Int. Congr. Math. (Cambridge, Massachusetts, August 30–September 6, 1950), Am. Math. Soc. (1952), pp. 216–223.
Yu. G. Lumiste, “Semisymmetric submanifolds,” J. Math. Sci., 70, No. 2 (1994), pp. 1609–1623.
J. Mikeš, Geodesic mappings of semisymmetric spaces, preprint VINITI No. 3924-76Dep (1976).
J. Mikeš, “Geodesic mappings onto semisymmetric space,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 2 (1994), pp. 37-43.
J. Mikeš, “Geodesic mappings of affine-connected and Riemannian spaces,” J. Math. Sci., 78 (1996), pp. 311–333.
J. Mikeš, V. E. Berezovskii, E. Stepanova, and H. Chudá, “Geodesic mappings and their generalizations,” J. Math. Sci., 217 (2016), pp. 607–623.
J. Mikeš and M. Chodorová, “On concircular and torse-forming vector fields on compact manifolds,” Acta Math. Acad. Paedagog. Nyházi., 26 (2010), pp. 329–335.
J. Mikeš, A. Vanžurová, and I. Hinterleitner, Geodesic Mappings and Some Generalizations, Palacky Univ. Press, Olomouc (2009).
J. Mikeš et al., Differential Geometry of Special Mappings, Palacky Univ. Press, Olomouc (2015).
V. A. Mirzoyan, “Classification of Ric-semiparallel hypersurfaces in Euclidean spaces,” Mat. Sb., 191, No. 9 (2000), pp. 65-80.
M. Najdanović, M. Zlatanović, and I. Hinterleitner, “Conformal and geodesic mappings of generalized equidistant spaces,” Publ. Inst. Math. Beograd., 98 (112) (2015), pp. 71–84.
K. Nomizu, “On hypersurfaces satisfying a certain condition on the curvature tensor,” Tôhoku Math. J., 20 (1968), pp. 46–59.
A. P. Norden, Spaces of Affine Connection [in Russian], Nauka, Moscow (1976).
A. Z. Petrov, New Methods in General Relativity [in Russian], Nauka, Moscow (1966).
H. S. Ruse, “On simply harmonic spaces,” J. London Math. Soc., 21 (1947), pp. 243–247.
A. A. Sabykanov, J. Mikeš, and P. Peška, “On semisymmetric projectively Euclidean spaces,” in: Proc Int. Conf. “Modern Geometry and Its Applications” (Kazan, Novermber 27 – December 3, 2017), Kazan Federal Univ., Kazan (2017), pp. 123–125.
A. Sabykanov, J.Mikeš, and P. Peška, “Recurrent equiaffine projective-Euclidean spaces,” Filomat, 33, No. 4 (2019).
K. Sekigawa, “On some hypersurfaces satisfying R(X, Y )∙R = 0,” Tensor, 25 (1972), pp. 133–136.
Shirokov P. A., “Constant fields of vectors and tensors in Riemannian spaces,” Izv. Kazan. Fiz.- Mat. Obshch., No. 25 (1925), pp. 86–114.
Shirokov P. A., Selected Works in Geometry [in Russian], Kazan (1966).
N. S. Sinyukov, “On geodesic mappings of Riemannian spaces on symmetric Riemannian spaces,” Dokl. Akad. Nauk SSSR, 98, No. 1 (1954), pp. 21–23.
N. S. Sinyukov, Geodesic Mappings of Riemannian spaces [in Russian], Nauka, Moscow (1979).
N. S. Sinyukov, “Almost geodesic mappings of affinely connected and Riemannian spaces,” Itogi Nauki Tekhn. Ser. Probl. Geom., 13 (1982), pp. 3-26.
M. Stanković, S. Mincić, Lj. Velimirović, and M. Zlatanović, “On equitorsion geodesic mappings of general affine connection spaces,” Rend. Semin. Mat. Univ. Padova, 124 (2010), pp. 77–90.
S. Stepanov, I. Shandra, and J. Mikeš, “Harmonic and projective diffeomorphisms,” J. Math. Sci., 207 (2015), pp. 658–668.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 179, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, 2020.
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Sabykanov, A.S., Mikeš, J. & Peška, P. Symmetric, Semisymmetric, and Recurrent Projectively Euclidean Spaces. J Math Sci 276, 410–416 (2023). https://doi.org/10.1007/s10958-023-06757-8
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DOI: https://doi.org/10.1007/s10958-023-06757-8