Cayley hypersurfaces

Abstract

We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.

Lorentzian Affine Hypersurfaces with Parallel Cubic Form

“…positive) mean curvature, whereas (vii) and (viii) are hyperbolic affine spheres. The hypersurface (ii) is known as the 3-dimensional Cayley hypersurface, which is an extended notion of the classical Cayley surface, constructed by Eastwood and Ezhov [6] for each higher dimension. Moreover, (iv), (v) and (viii) are affine spheres with constant sectional curvature, the later were completely classified in [18] for the locally strongly convex case and [17] for the genenral case with non-vanishing Pick invariant.…”

A Characterization of Cayley Hypersurface and Eastwood and Ezhov Conjecture

“…On the other hand, Eastwood and Ezhov generalized the Cayley surface with a somewhat different view point in [7]. Considering the properties of the automorphism group of the Cayley surface, they generalized and constructed a Cayley hypersurface which is homogeneous and unique up to affine congruence in each dimension.…”

A differential geometric characterization of the Cayley hypersurface

“…Proposition ( [7] Eastwood and Ezhov [7] further conjectured that the last three properties of the Proposition should characterize the Cayley hypersurface. This conjecture was proved by Choi and Kim [2] under the additional condition that the domain in R n+1 , being bounded by the hypersurface which is expressed as a graph of a function defined on R n , is affine homogeneous.…”

“…In [13], Nomizu and Pinkall showed that the Cayley surface can be characterized, up to an affine congruence, as the unique affine surface whose cubic form is non-zero and is parallel relative to the induced affine connection, i.e., C = 0 and ∇C = 0. For general dimensions, Eastwood and Ezhov [7] exhibited a homogeneous hypersurface in R n+1 , named as Cayley hypersurface, defined by…”

“…This conjecture was proved by Choi and Kim [2] under the additional condition that the domain in R n+1 , being bounded by the hypersurface which is expressed as a graph of a function defined on R n , is affine homogeneous. On the other hand, Eastwood and Ezhov [7] raised an interesting question as follows:…”