The Magid-Ryan conjecture for \(4\)-dimensional affine spheres
A conjecture of \textit{M. Magid} and \textit{P. Ryan} [Trans. Am. Math. Soc. 330, 887-901 (1992; Zbl 0765.53009)] says that an affine hypersphere \(M^n\) in \(\mathbb{R}^{n+ 1}\) with constant sectional curvature \(a\) and with nonzero Pick invariant \(J\) is affinely equivalent with \((x^2_1\pm x^2_2)(x^2_3\pm x^2_4)\cdots(x^2_{2m- 1}\pm x^2_{2m})= 1\) (\(n+1= 2m\)) or with \((x^2_1\pm x^2_2)(x^2_3\pm x^2_4)\cdots (x^2_{2m- 1}\pm x^2_{2m}) x_{2m+ 1}= 1\) (\(n+1= 2m+1\)) while the curvature \(a\) vanishes. After this conjecture has been verified by Li, Penn, Simon, Magid, Yu, Ryan, Kriele and Vrancken in the cases of dimension \(n=2\) or \(n=3\) and in the cases of a positive definite or Lorentzian affine metric, the authors now show that it is also correct for dimension \(n=4\). Their proof consists of the choice of an appropriate local basis for the tangent bundle of \(M^4\).
scientific article; zbMATH DE number 1416700