Abstract
In this paper, we study a whole family of n-dimensional equiaffine homogeneous hypersurfaces with a parameter α, constructed by Eastwood and Ezhov (Proc Steklov Inst Math 253:221–224, 2006), called the generalized Cayley hypersurfaces. By introducing a new parametrization we find that the generalized Cayley hypersurfaces are improper affine hypersphere with flat affine metric and vanishing Pick invariant, whose difference tensor K satisfies \({\nabla^{(\alpha)}K=0 \,\,{\rm and}\,\, K^{n-1} \neq 0}\), where the affine \({\alpha{\rm -connection}\,\, \nabla^{(\alpha)}}\) of information geometry is first introduced on affine hypersurface for each \({\alpha\in\mathbb{R}}\). As main result, we establish a characterization of the generalized Cayley hypersurfaces by the last two properties for some nonzero constant α.
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This research was supported by the Grants of NSFC-11326072 and NSFC-11401173.
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Li, C., Zhang, D. The Generalized Cayley Hypersurfaces and Their Geometrical Characterization. Results. Math. 68, 25–44 (2015). https://doi.org/10.1007/s00025-014-0419-x
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DOI: https://doi.org/10.1007/s00025-014-0419-x