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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hypersurfaces of order two


Author: Tibor Bisztriczky
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
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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].


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DOI: https://doi.org/10.1090/S0002-9947-1976-0405319-7
Keywords: Index, degree of degeneracy, tangent hyperplanes, boundaries of linearly connected sets, ruled quadrics
Article copyright: © Copyright 1976 American Mathematical Society