Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Lorentzian affine hyperspheres
with constant affine sectional curvature


Authors: Marcus Kriele and Luc Vrancken
Journal: Trans. Amer. Math. Soc. 352 (2000), 1581-1599
MSC (1991): Primary 53A15
DOI: https://doi.org/10.1090/S0002-9947-99-02379-X
Published electronically: July 26, 1999
MathSciNet review: 1621765
Full-text PDF Free Access

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 [Enhancements On Off] (What's this?)

  • [DMV] F. Dillen, M. Magid and L. Vrancken, Affine hyperspheres with constant affine sectional curvature, preprint.
  • [LP] A.M. Li and G. Penn, Uniqueness theorems in affine differential geometry, Res. Math. 13 (1988), 308-317. MR 89h:53033
  • [LSZ] A.M. Li, U. Simon and G. Zhao, Global affine differential geometry of hypersurfaces, De Gruyter, Berlin, 1993. MR 95e:53016
  • [MR1] M. Magid and P. Ryan, Flat affine spheres $\mathbf{R}^3$, Geom. Dedicata 33 (1990), 277-288. MR 91e:53016
  • [MR2] -, Affine 3-spheres with constant affine curvature, Trans. Amer. Math. Soc. 330 (1992), 887-901. MR 92f:53063
  • [NS] K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge University Press, Cambridge, 1994. MR 96e:53014
  • [NV] K. Nomizu and L. Vrancken, Geodesics in affine differential Geometry, International J. Mathematics 6 (1995), 749-766. MR 96f:53020
  • [R] J. Radon, Zur Affingeometrie der Regelflächen, Leipziger Berichte 70 (1918), 147-155.
  • [S] U. Simon, Local classification of two-dimensional affine spheres with constant curvature metric, Differential Geom. Appl. 1 (1991), 123-132. MR 94g:53006
  • [VLS] L. Vrancken, A. M. Li and U. Simon, Affine spheres with constant affine sectional curvature, Math. Z. 206 (1991), 651-658. MR 92m:53099
  • [W] C. Wang, Canonical equiaffine hypersurfaces in $\mathbb{R}^{n+1}$, Math Z. 214 (1993), 579-592. MR 95e:53079

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53A15

Retrieve articles in all journals with MSC (1991): 53A15


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: https://doi.org/10.1090/S0002-9947-99-02379-X
Received by editor(s): July 10, 1997
Received by editor(s) in revised form: April 1, 1998
Published electronically: 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
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society