Hypersurfaces of order two
HTML articles powered by AMS MathViewer
- by Tibor Bisztriczky PDF
- Trans. Amer. Math. Soc. 220 (1976), 205-233 Request permission
Abstract:
A hypersurface ${S^{n - 1}}$ of order two in the real projective n-space is met by every straight line in maximally two points; cf. [1, p. 391]. We develop a synthetic theory of these hypersurfaces inductively, basing it upon a concept of differentiability. We define the index and the degree of degeneracy of an ${S^{n - 1}}$ and classify the ${S^{n - 1}}$ in terms of these two quantities. Our main results are (i) the reduction of the theory of the ${S^{n - 1}}$ to the nondegenerate case; (ii) the Theorem (A.5.11) that a nondegenerate ${S^{n - 1}}$ of positive index must be a quadric and (iii) a comparison of our theory with Marchaud’s discussion of “linearly connected” sets; cf. [3].References
- Otto Haupt and Hermann Künneth, Geometrische Ordnungen, Die Grundlehren der mathematischen Wissenschaften, Band 133, Springer-Verlag, Berlin-New York, 1967 (German). MR 0227907 A. Marchaud, Les surfaces du second ordre en géométrie finie, J. Math. Pures Appl. 15 (1936), 293-300.
- André Marchaud, Sur les ensembles linéairement connexes, Ann. Mat. Pura Appl. (4) 56 (1961), 131–157 (French). MR 170248, DOI 10.1007/BF02414269
- Ralph Park, Topics in direct differential geometry, Canadian J. Math. 24 (1972), 98–148. MR 293563, DOI 10.4153/CJM-1972-012-1 P. Scherk, Über differenzierbare Kurven und Bögen. I. Zum Begriff der Charakteristik, Časopis Pěst. Mat. Fys. 66 (1937), 165-171.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 220 (1976), 205-233
- MSC: Primary 53C75
- DOI: https://doi.org/10.1090/S0002-9947-1976-0405319-7
- MathSciNet review: 0405319