Lorentzian affine hyperspheres with constant affine sectional curvature

Kriele, Marcus; Vrancken, Luc

doi:10.1090/S0002-9947-99-02379-X

Lorentzian affine hyperspheres with constant affine sectional curvature
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by Marcus Kriele and Luc Vrancken PDF

Trans. Amer. Math. Soc. 352 (2000), 1581-1599 Request permission

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 \[ (x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2}) = 1\] or \[ (x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2})x_{2m+1} = 1\] 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.

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