On the correspondence between symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds

Abstract

In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the classification theorem for the locally strongly convex equiaffine hypersurfaces with parallel Fubini-Pick forms, which has been established recently by Z.J. Hu etc in a totally different way.

A New Characterization of Calabi Composition of Hyperbolic Affine Hyperspheres

“…Corollary 3.2. ( [19]) A locally strongly convex and affine symmetric hypersurface x : M n → R n+1 is locally affine equivalent to the Calabi composition of some hyperbolic affine hyperspheres possibly including point factors if and only if M n is reducible as a Riemannian manifold with respect to the affine metric.…”

On the Equiaffine Symmetric Hyperspheres

“…Note that for a given locally strongly convex hypersurface x : M n → R n+1 with the affine metric g, (M n , g) is a Riemannian manifold. Then we have the following characterization of Calabi composition of symmetric factors which is important in the proof of Theorem 1.1: [24]) A locally strongly convex and affine symmetric hypersphere x : M n → R n+1 is locally affine equivalent to the Calabi composition of some hyperbolic affine hyperspheres possibly including point factors if and only if M n is reducible as a Riemannian manifold with respect to the affine metric.…”

“…≡{A ∈ E 6(−26) ; A(I 3 ) = I 3 }. Similar to the above, one can perform a computation which shows that the trace of an arbitrary element of e 6(−26) must vanish (for the detail, see [24]). Thus we have This shows that x is an immersion at o and thus is an immersion globally since x is equivariant.…”

“…In this paper, on the basis of a recent characterization of Calabi composition of hyperbolic hypersphere ( [24], [25]), we make use of the idea by H. Naitoh in [26] for classification of totally real parallel submanifolds in the complex projective space, to provide a direct proof of the complete classification of symmetric affine hyperspheres. Then, via an earlier result of the author, we easily give an alternative and simpler proof for the classification theorem (Theorem 4.6) for the affine hypersurface with parallel Fubini-Pick forms, which has already been established by Z.J.…”

A classification theorem of nondegenerate equiaffine symmetric hypersurfaces

“…In fact, various kind of affine hyperspheres with particular properties are widely studied. For example, the proof of the Calabi's conjecture on hyperbolic hyperspheres and the rigidity theorems on the ellipsoid in the locally strongly convex case (see for example, [30], [31] and [32]); the classification of hyperspheres of constant sectional affine curvatures ( [53], [54] and [29]); the study of homogeneous hyperspheres ( [49], [50], [51] for the relation with symmetric cones, [8] for the Calabi-type composition), the characterization of the Calabi composition of hyperbolic hyperspheres ([17]; also [37] and [38] in a different manner); the isotropic affine hyperspheres ( [3]); and so on.…”