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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On projective spheres and Fubini and Tzitzeica-Wilczyński pseudospheres
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by Froim Marcus PDF
Proc. Amer. Math. Soc. 93 (1985), 512-520 Request permission

Abstract:

The definition of a projective pseudosphere is given, and the following results are proved: ( 1 ) There exists one and only one Fubini pseudosphere iff \[ L = M = - \frac {{3{\varphi ^2}}}{2},\quad \beta = \gamma = \varphi ;\quad \varphi = \varphi (\tau )\quad (\tau = u + \upsilon ).\] (2) There exist two classes of limiting Tzitzeica- Wilczynski pseudospheres or improper affine spheres. These pseudospheres admit, beside the first directrix or affine normal, the first Fubini principal straight line or the conjugate of the first directrix with respect to the canonical tangent and projective normal. The asymptotes of the pseudosphere are twisted cubics.
References
    G. Fubini and E. Čech, Geometria proiettiva differenziale, vol. 1, Zanichelli, Bologna, 1926. —, Introduction à la géométrie projective differentielle des surfaces, Gauthier-Villars, Paris, 1931. E. Y. Wilczynski, Über Flächen mit unbestimten Direktrixkurven, Math. Ann. 76 (1915), 129-160. W. Blaschke, Differential geometrie, Springer-Verlag, Berlin, 1923, pp. 212, 216. E. Cartan, Sur la connexion projective des surfaces, C. R. Acad. Sci. Paris Ser. A-B 178 (1929), 750. J. Kaucky, Étude des surfaces dont une droite canonique passe par un point fixe, Publ. No. 109 de la Faculté des Sci. de l’Université Masaryk, Brno. —, Sur les surfaces dont une droite canonique passe par un point fixe, Rend. Accad. Leincei 9 (1929).
  • Froim Marcus, On the results of I. Kaucky concerning the problem of the determination of surfaces for which a canonical line passes through a fixed point, Acad. Roy. Belg. Bull. Cl. Sci. (5) 63 (1977), no. 3, 274–282 (English, with French summary). MR 487824
  • Froim Marcus, Completion of Čech’s and Kaucký’s results on the analysis of surfaces on which a canonical line passes through a fixed point, Ann. Mat. Pura Appl. (4) 114 (1977), 319–330. MR 487825, DOI 10.1007/BF02413793
  • Froim Marcus, Sur les surfaces dont deux normales ou droites canoniques passent par un point fixe, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 24 (1978), no. 2, 345–349 (French). MR 533762
  • F. Marcus, Sur les réseaux de Koenigs, Rev. Math. Pures Appl. 2 (1957), 555–559 (French). MR 95491
    • E. Kamke, Differentialgleichungen 3 Auflage, Chelsea, New York, pp. 26-28.
  • Froim Marcus, Again on the surfaces which allow $\infty ^{2}$ projective transformations into themselves, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 22 (1976), no. 1, 35–48. MR 642205
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 93 (1985), 512-520
  • MSC: Primary 53A20
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0774015-X
  • MathSciNet review: 774015