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A classification of isotropic affine hyperspheres

Abstract : We study affine hypersurfaces M, which have isotropic difference tensor. Note that, any surface always has isotropic difference tensor. In case that the metric is positive definite, such hypersurfaces have been previously studied in [O. Birembaux and M. Djoric, Isotropic affine spheres, Acta Math. Sinica 28(10) 1955-1972.] and [O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417(2) (2014) 918-962.] We first show that the dimension of an isotropic affine hypersurface is either 5, 8, 14 or 26. Next, we assume that M is an affine hypersphere and we obtain for each of the possible dimensions a complete classification.
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Marilena Moruz, Luc Vrancken. A classification of isotropic affine hyperspheres. International Journal of Mathematics, World Scientific Publishing, 2016, 27 (9), pp.1650074. ⟨10.1142/S0129167X16500749⟩. ⟨hal-03173175⟩

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