Abstract
The theory of F-planar maps of Riemannian spaces and affinely connected spaces developed by J. Mikeš and N. S. Sinyukov [1–6] naturally extends the theory of geodesic and holomorphic projective maps. In the present paper we find basic equations of infinitesimal F-planar maps and study these equations. The F-planar maps are maps between spaces endowed with affinor structures. The geometry of Riemannian spaces and affinely connected spaces endowed by affinor structures was investigated by A. P. Shirokov (see, e.g., [7–14]) who also studied maps between spaces of this type ([13, 14]).
Similar content being viewed by others
References
J. Mikeš and N. S. Sinyukov, “On quasiplanar Mappings of Spaces of Afflne Connection,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 1, 55–61 (1983) [Soviet Mathematics (Iz. VUZ) 27, (1), 53–59 (1983)].
J. Mikeš, “F-Planar Mappings and Transformations,” in Proceedings of the Conference on Differential Geometry and its Applications (August 24–30, 1986, Brno, Czechoslovakia), pp. 245–254.
J. Mikeš, “On F-Planar and f-Planar Maps, Transformations, and Deformations,” Geometry of Generalized Spaces, Penza, pp. 60–65 (1992).
J. Mikeš, “On Special F-Planar Maps of Affinely Connected Spaces,” Vestnik Mosk. Univ., No. 3, 18–24 (1994).
J. Mikeš, “Holomorphically Projective Mappings and their Generalizations,” J. Math. Sci. 89(3), 1334–1353 (1998).
I. Hinterleitner and J. Mikes, “On F-Planar Mappings of Spaces with Affine Connections,” Note Mat. 27(1), 111–118 (2007).
A. P. Shirokov, “On A-Spaces,” in 125th Anniversary on Lobachevskii Non-Euclidean Geometry 1826–1951 (Moscow-Leningrad, GITTL, 1952), pp. 195–200.
A. P. Shirokov, “On a Property of Covariant Constant Affinors,” Sov. Phys. Dokl. 102, 461–464 (1955).
A. P. Shirokov, “Structures on Differentiable Manifolds,” in Itogi nauki i Tekhn. Ser. Algebra. Tolologiya. Geometriya (VINITI, Moscow, 1967 (1969)), pp. 127–188.
A. P. Shirokov, “Spaces over Algebras and their Applications,” in Itogi nauki i tekhn. Ser. Sovremen. Matem. Prilozh. Tem. Obzor. 73 (VINITI, Moscow, 2002), pp. 135–161.
V. V. Vishnevskii, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras (Kazan Univ. Press, Kazan, 1985) [in Russian].
L. E. Evtushik, Yu. G. Lümiste, N. M. Ostianu, and A. P. Shirokov, “Differential Geometric Structures on Manifolds,” in Itogi nauki i tekn. Ser. Problemy Geometrii (VINITI, Moscow, 1979), 246 p.
K. M. Egiazarjan and A. P. Shirokov, “Projection of Connections in Fiber Bundles and its Application to the Geometry of Spaces Over Algebras,” Differents. Geometriya 4, 132–140 (1979).
N. V. Talantova and A. P. Shirokov, “Projective Models of Unitary Spaces of Constant Curvature Over Algebra of Dual Numbers,” Trudy Geometrich. Semin. 16, 103–110 (1984).
A. Z. Petrov, “Simulation of Physical Fields,” in Gravitatsiya i Teoriya Otnositel’nosti, Kazan, Nos. 4–5, 7–21 (1968).
L. P. Eisenhart, Riemannian Geometry (Inost. Lit., Moscow, 1948) [Russian translation].
M. L. Gavrilchenko, V. A. Kiosak, and J. Mikeš, “Geodesic Deformations of Hypersurfaces of Riemannian Spaces,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 23–29 (2004) [Russian Mathematics (Iz. VUZ) 48 (11), 20–26 (2004)].
Zh. Radulovich, J. Mikeš, and M. L. Gavrilchenko, Geodesic Mappings and Deformations of Riemannian Spaces (Izd. CID, Podgorica, Izd. OGU, Odessa, 1997).
J. Mikeš, “Geodesic Mappings of Affine-Connected and Riemannian Spaces,” J. Math. Sci. 78(3), 311–333 (1996).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I. Hinterleitner, J. Mikeš, J. Stránská, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 4, pp. 16–22.
About this article
Cite this article
Hinterleitner, I., Mikeš, J. & Stránská, J. Infinitesimal F-planar transformations. Russ Math. 52, 13–18 (2008). https://doi.org/10.3103/S1066369X08040026
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X08040026