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 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Lorentzian affine hyperspheres with constant affine sectional curvature

Author(s): Marcus Kriele; Luc Vrancken
Journal: Trans. Amer. Math. Soc. 352 (2000), 1581-1599.
MSC (1991): Primary 53A15
Posted: July 26, 1999
MathSciNet review: 1621765
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Abstract | References | Similar articles | Additional information

Abstract: We study affine hyperspheres $M$ with constant sectional curvature (with respect to the affine metric $h$). A conjecture by M. Magid and P. Ryan states that every such affine hypersphere with nonzero Pick invariant is affinely equivalent to either

\begin{displaymath}(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2}) = 1\end{displaymath}

or

\begin{displaymath}(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2})x_{2m+1} = 1\end{displaymath}

where the dimension $n$ satisfies $n=2m-1$ or $n=2m$. Up to now, this conjecture was proved if $M$ is positive definite or if $M$ is a $3$-dimensional Lorentz space. In this paper, we give an affirmative answer to this conjecture for arbitrary dimensional Lorentzian affine hyperspheres.

References:

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F. Dillen, M. Magid and L. Vrancken, Affine hyperspheres with constant affine sectional curvature, preprint.

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A.M. Li and G. Penn, Uniqueness theorems in affine differential geometry, Res. Math. 13 (1988), 308-317. MR 89h:53033

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A.M. Li, U. Simon and G. Zhao, Global affine differential geometry of hypersurfaces, De Gruyter, Berlin, 1993. MR 95e:53016

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M. Magid and P. Ryan, Flat affine spheres $\mathbf{R}^3$, Geom. Dedicata 33 (1990), 277-288. MR 91e:53016

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-, Affine 3-spheres with constant affine curvature, Trans. Amer. Math. Soc. 330 (1992), 887-901. MR 92f:53063

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Additional Information:

Marcus Kriele
Affiliation: Technische Universität Berlin, Fachbereich Mathematik MA 8-3, Strasse des 17 Juni 135, D-10623 Berlin, Germany
Email: kriele@sfb288.math.tu-berlin.de

Luc Vrancken
Affiliation: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
Address at time of publication: Technische Universität Berlin, Fachbereich Mathematik, Sekr. MA8-3, Strasse des 17 Juni 135, D-10623 Berlin, Germany
Email: luc@sfb288.math.tu-berlin.de, luc@sfb288.math.tu-berlin.de

DOI: 10.1090/S0002-9947-99-02379-X
PII: S 0002-9947(99)02379-X
Received by editor(s): July 10, 1997
Received by editor(s) in revised form: April 1, 1998
Posted: July 26, 1999
Additional Notes: The first author was supported by a Research Fellowship of the Research Council of the K.U. Leuven
Research supported by the grant OT/TBA/95/9 of the Research Council of the Katholieke Universiteit Leuven.
The authors would like to thank the referee for improving some arguments in the paper.
Dedicated: Dedicated to the sixtieth birthday of Udo Simon
Copyright of article: Copyright 2000, American Mathematical Society




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