Abstract
In the paper we show that mappings of Grassmann spaces sending base subsets to base subsets are induced by strong embeddings of the corresponding projective spaces.
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References
Baer R., Projective Geometry and Linear Algebra, Academic Press, New York, 1952.
Benz W., Geometrische Transformationen. B.I. Wissenschaftsverlag, Mannheim Leipzig Wien Zürich, 1992.
Benz W., Real Geometries., B.I. Wissenschaftsverlag, Mannheim Leipzig Wien Zürich, 1994.
Bertini E., Einfuehrung in die projektive Geometrie mehr-dimensionaler Raeume., Vienna, 1924.
Brauner H., Über die von Kollineationen projectiver Räume induzierten Geradenabbildungen, Sitz. Ber. Akad. Wiss., math-naturw. Kl. Abt. II, 197 (1988), 327–332.
Brezuleanu A., Rădulescu D. C, About full or injective lineations, J. Geom. 23 (1984), 45–60.
Ceccherini P. V. Collineazioni e semicollineazioni tra spazi affini o proiettivi, Rend. Mat. Roma (5) 26 (1967), 309–348.
Chow W. L., On the geometry of algebraic homogeneous spaces, Ann. of Math., 50 (1949), 32–67.
Faure C. A., Frölicher A., Morphisms of Projective Geometries and Semilinear maps, Geom. Dedicata, 53 (1994), 237–262.
Faure C. A., An Elementary Proof of the Fundamental Theorem of Projective Geometry, Geom. Dedicata, 90 (2002), 145–151
Havlicek H., A Generalization of Brauner’s Theorem on Linear Mappings, Mitt. Math. Sem. Univ. Gieβen, 215 (1994), 27–41.
Havlicek H., On Isomorphisms of Grassmann Spaces, Mitt. Math. Ges. Hamburg. 14 (1995), 117–120.
Huang W.-L., Adjacency preserving transformations of Grassmann spaces, Abh. Math. Sem. Univ. Hamburg 68 (1998), 65–77.
Huang W.-L., Kreuzer A., Basis preserving maps of linear spaces, Arch. Math. (Basel) 64(1995), 6, 530–533.
Karzel H., Kroll H. J., Geschichte der Geometrie seit Hilbert, Wissenschaftliche Buchgesellschaft, Darmstadt, 1988.
Kreuzer A., On the definition of isomorphisms of linear spaces, Geom. Dedicata, 61(1996), 279–283.
Kreuzer A., On isomorphisms of Grassmann spaces, Aequationes Math. 56(1998) n. 3, 243–250.
Lester J. A., Distance preserving transformations, Hand book of incidence geometry. Buildings and foundations, Edited by F. Buekenhout, North-Holland, Amsterdam, 1995, 921–944.
Pankov M., Projections of k-dimensional subsets of {ℝsun} onto k-dimensional planes and irregular subsets of the Grassmannian manifolds, Topol. App. 101(2000), 121–135.
Pankov M., Irregular subsets of the Grassmannian manifolds and their mappings, Mat. Fiz., Anal., Geom., 7(2000), 3, 331–344.
Pankov M., Transformations of Grassmannians and automorphisms of classical groups, J. Geom. 75(2002), 132–150.
Pankov M., Transformations of Grassmannians preserving the class of base sets, J. Geom. 79(2004), 169–176.
Veblen O., Collineations in a finite projective geometry, Trans. Amer. Math. Soc. 8 (1907), 366–368.
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Pankov, M. A characterization of geometrical mappings of Grassmann spaces. Results. Math. 45, 319–327 (2004). https://doi.org/10.1007/BF03323386
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DOI: https://doi.org/10.1007/BF03323386