Abstract
A review of the current state of the diametral theory of algebraic hypersurfaces in the real Euclidean space is given.
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V. F. Ignatenko, “Infinite transparent groups generated by skew reflections,” in: Abstracts of the International Conference on Geometry “as a Whole” [in Russian], Cherkassy (1995), pp. 33–34.
V. F. Ignatenko, “Rebuilding method and wild groups of skew symmetries,” in: Abstracts of the Second Crimean Mathematical School “Method of Lyapunov Functions and Its Applications” [in Russian], Simferopol, (1995), pp. 24–25.
E. B. Vinberg and V. L. Popov, “Theory of invariants,” in: VINITI Series in Contemporary Problems of Mathematics. Fundamental Trends [in Russian], Vol. 55, VINITI, Moscow (1989), pp. 137–309.
V. F. Ignatenko, “On the geometric theory of invariants of groups generated by reflections,” in: VINITI Series in Problems of Geometry [in Russian], Vol. 21, VINITI, Moscow (1989), pp. 155–208.
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer (1986).
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).
P. I. Golod and A. U. Klimyk, Mathematical Foundations of the Theory of Symmetries [in Ukrainian], Naukova Dumka, Kiev (1992).
M. Berger, Geometry, Springer (1987).
V. I. Arnol’d, “Topological problems of the theory of wave propagation,” Usp. Mat. Nauk, 51, No. 1, 3–50 (1996).
V. F. Ignatenko, “Some problems of the geometric theory of invariants of groups generated by orthogonal and skew reflections,” in: VINITI Series in Problems of Geometry [in Russian], Vol. 16, VINITI, Moscow (1984), pp. 195–229.
A. S. Smogorzhevskii and E. S. Stolova, A Handbook in the Theory of Plane Curves of the Third Order [in Russian], Fizmatgiz, Moscow (1961).
Yu. A. Aminov, “Torsion of two-dimensional surfaces in Euclidean spaces,” Ukr. Geom. Sb., Issue 17, 3–14 (1975).
A. P. Fomenko, “Dupin indicatrix of a two-dimensional surface in E4,” in: Abstracts of the International Conference on Geometry “as a Whole” [in Russian], Cherkassy (1995), p. 93.
A. A. Borisenko and Yu. A. Nikolaevskii, “Classification of points of three-dimensional surfaces by the Grassmann image,” Ukr. Geom. Sb., Issue 32, 11–27 (1989).
V. F. Ignatenko, “On some classes of algebraic surfaces with infinite set of planes of skew symmetry,” Dinam. Sist., 8, 119–126 (1989).
V. F. Ignatenko, “Geometry of algebraic surfaces with symmetries,” in: VINITI Series in Problems of Geometry [in Russian], Vol. 11, VINITI, Moscow (1980), pp. 203–240.
B. A. Rosina, “Sulla teoria diametrale delle superficile algebriche,” Bull. Soc. Roy. Sci. Liége, 31, Nos. 3-4, 146–157 (1962).
V. F. Ignatenko, “On the estimate of the number of principal diametral planes of an algebraic surface of the space E4,” Ukr. Geom. Sb., Issue 11, 31–35 (1971).
L. Flatto, “Invariants of finite reflection groups,” Enseign. Math., 24, Nos. 3-4, 237–292 (1978).
S. P. Sopov, “On one class of uniform polyhedrons,” Ukr. Geom. Sb., Issue 7, 130–140 (1970).
A. I. Medyanik, “Regular simplex inscribed in a cube,” Ukr. Geom. Sb., Issue 13, 109–112 (1973).
V. F. Ignatenko, “On diametral planes and planes of skew symmetry of an algebraic surface of the space E m ” Ukr. Geom. Sb., Issue 20, 35–46 (1977).
B. A. Rosina, “Classificazione delle curve quartiche plane secondo ta teoria diametrale delle curve algebriche piane,” Ann. Univ. Ferrara. Sez. VII, 21, 1–15 (1975).
B. A. Rosina, “Classificazione delle curve algebriche piane di ordine qualunque secondo la teoria delle curve algebriche piane” Atti Acad. Sci. Lett, ed Ari Palermo. Sez. IV, part I, 34, No. 2, 101–121 (1976).
O. I. Rudnitskii, “On one property of a degenerate diametral surface of an algebraic surface of the space E m“Dinam. Sist, 13, 131–135 (1994).
V. F. Ignatenko, “On some problems of the geometric theory of invariants of groups generated by reflections,” in: Differential-Geo-metric Structures on Manifolds and Their Applications [in Russian], Chernovtsy (1991), pp. 230–234, dep. VINITI No. 562–V91.
S. S. Bushgens Differential Geometry [Russian translation], Gostekhizdat, Moscow-Leningrad (1940).
V. F. Ignatenko, “Invariants of finite and infinite groups generated by reflections,” J. Math. Socil., 78, No. 3, 334–361 (1972).
A. V. Pogorelov, “Improper convex affine hyperspheres,” Dokl. Akad. Nauk SSSR, 202, No. 5, 1008–1011 (1972).
A. V. Pogorelov, Multidimensional Minkowski Problem [in Russian], Nauka, Moscow (1975).
A. D. Milka, “Indecomposability of a convex surface,” Ukr. Geom. Sb., Issue 13, 112–129 (1973).
Ya. P. Blank, “Conjugate nets of conic lines,” Dokl. Akad. Nauk SSSR, 14, No. 6, 755–758 (1949).
B. A. Rosina, “Sulle superficie algebriche di ordine 2n con und conica (almero) doppia all’infinito (quadriche generalizzate),” Ann. Univ. Ferrara. Sez. VII, 2, 141–149 (1953).
V. F. Ignatenko, “On the geometry of a generalized quadric,” Dinam, Sist., 13, 127–131 (1994).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 639–653, May, 1998.
The present paper is based on talks delivered by the author at the International Conference on Geometry “as a Whole” [1] and the Second Crimean Mathematical School “Method of Lyapunov Functions and Its Applications” [2].
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Ignatenko, V.F. Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. I. Ukr Math J 50, 726–740 (1998). https://doi.org/10.1007/BF02514326
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DOI: https://doi.org/10.1007/BF02514326