Abstract
In the present paper we note some of the most brilliant and natural applications of the theory of affine connection spaces to projective-differential geometry in the method of normalization of A. P. Norden, and we also consider a number of questions connected with the theory of compositions.
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Literature cited
K. M. Egiazaryan, “Projection of invariant connections on principal fiber spaces,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 97–101 (1978).
K. M. Egiazaryan, “Almost product affine connection manifolds,” Tr. Geometr. Sem. Kazan. Univ., No. 11, 29–55 (1979).
K. M. Egiazaryan, “Affine connections on Grassman manifolds,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 76–78 (1980).
K. M. Egiazaryan, “Projected invariant affine connections,” Tr. Geometr. Sem. Kazan. Univ., No. 12, 27–37 (1980).
K. M. Egiazaryan, “Affine connections on a real model of a Grassman manifold over an algebra,” Tr. Geometr. Sem. Kazan. Univ., No. 13, 12–20 (1981).
K. M. Egiazaryan and A. P. Shirokov, “Projected connections in bundles and their application to the geometry of spaces over algebras,” Differents. Geometriya (Saratov), No. 4, 132–140 (1979).
E. Cartan, Affine, Projective, and Conformal Connection Spaces [Russian translation], Kazan. Univ. (1962).
E. M. Kuznetsova, “Affine connections on manifolds of m-pairs of n-dimensional projective space,” [in Russian], Kazan. Univ. (1983) (manuscript deposited in VINITI, July 14, 1983, No. 3951-83 Dep.).
E. M. Kuznetsova, “Affine connections on manifolds of pairs of points of one-dimensional projective space P1 over the algebraU,” [in Russian], Kazan Univ. (1984) (manuscript deposited in VINITI, April 16, 1984, No. 2340-84 Dep.).
E. G. Neifel'd, “Affine connections on the normalized manifold of planes of projective space,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 11, 48–55 (1976).
E. G. Neifel'd, “Geometry of a surface in projective space over an algebra. Geometry of generalized spaces,” Ufa, 32–51 (1982).
E. G. Neifel'd, “Geometries defined by polarities in projective space over an algebra,” Tr. Geometr. Sem. Kazan. Univ., No. 15, 64–66 (1983).
E. G. Neifel'd, “Normalized family of planes and geometry of the second kind of a surface,” Tr. Geometr. Sem. Kazan. Univ., No. 16, 69–81 (1984).
A. P. Norden, Affine Connection Spaces [in Russian], Nauka, Moscow (1976).
A. P. Norden, “Structure of a connection on the manifold of lines of a non-Euclidean space,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 12, 84–94 (1972).
A. P. Norden, “Theory of compositions,” J. Sov. Math.,14, No. 3 (1980).
B. A. Rozenfel'd, “Unitary and fibered spaces,” Tr. Sem. Vektorn. Tenzorn. Anal., No. 7, 260–275 (1949).
B. A. Rozenfel'd, “Projective geometry as metric geometry,” Tr. Sem. Vektorn. Tenzorn. Anal., No. 8, 328–354 (1950).
B. A. Rozenfel'd, Symmetric spaces and their geometric application,” in: E. Cartan, Geometry of Lie Groups and Symmetric Spaces [Russian translation], Moscow (1949), pp. 331–368.
E. N. Sosov, “Remark on relative line-geometry,” Differents. Geometriya Mnogoobrazii Figur (Kaliningrad), No. 13, 91–94 (1982).
E. N. Sosov, “Relative line-geometry of affine space,” [in Russian], Kazan Univ., Kazan' (1984) (manuscript deposited in VINITI, June 1, 1984, No. 3619-84 Dep.).
L. Tuulmets, “Geometry of the homogeneous space of m-pairs and its manifolds,” Uch. Zap. Tartus. Univ., No. 464/22, 98–115 (1978).
B. N. Shapukov, “Connections on differentiable bundles,” J. Sov. Math.,29, No. 5 (1985).
A. P. Shirokov, “Normalizations in projective space with given fibration,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 216–221 (1974).
A. P. Shirokov, “Norden's method of normalization and real models of projective spaces over algebras,” Tr. Geometr. Sem. Kazan. Univ., No. 8, 145–152 (1975).
A. P. Shirokov, “Relative line-geometry,” Differents. Geometriya (Saratov), No. 3, 69–81 (1977).
V. A. Yur'ev, “Invariant connection on the manifold of lines of n-dimensional Euclidean space,” Sb. Aspirantsk. Rabot. Kazan. Univ. Tochn. Nauki, Mat., No. 1, 148–157 (1970).
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Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 17, pp. 131–151, 1985.
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Shirokov, A.P. Affine connection spaces (some aspects of the method of normalization of A. P. Norden). J Math Sci 37, 1239–1253 (1987). https://doi.org/10.1007/BF01091861
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DOI: https://doi.org/10.1007/BF01091861