A differential geometric characterization of the Cayley hypersurface

Abstract

Abstract. The so-called Cayley hypersurface, constructed by Eastwood and Ezhov, is a higher-dimensional extension of the classical Cayley surface. In this paper, we establish a differential geometric characterization of the Cayley hypersurface, which is an answer to Eastwood and Ezhov's question.

Locally homogeneous affine hyperspheres with constant sectional curvature

“…Proof. The proof of the first part is the same as the proof of Lemma 6.1 in [7], see also Lemma 3.2 in [16], although we obtain the local frame instead of basis. Note that Lemma 3.1 has been proved for n ¼ 3 (cf.…”

“…Without the condition of homogeneity, both the characterization of the Cayley hypersurface and that of the generalized Cayley hypersurfaces are obtained in [16] and [18], respectively.…”

Centro-affine hypersurface immersions with parallel cubic form

“…By (20) the form ζ defined by (17) is given by Proof. That M is locally a non-degenerate graph immersion follows from Lemmas 4.7 and 2.3.…”

“…2], they are improper affine hyperspheres [7,Prop. 4] with parallel cubic form and flat affine metric [20]. We now give the following description of the metrised Jordan algebras generated by the Cayley hypersurfaces.…”

“…The second hypersurface in (1) and the sixth hypersurface in (2) are generalizations of the Cayley surface known as Cayley hypersurfaces. These hypersurfaces were introduced in [7] and shown to be improper hyperspheres with vanishing Pick invariant, satisfying C = 0 and ∇C = 0 [20]. In [14,Sect.…”