A classification of isotropic affine hyperspheres (Q2822024)

For a nondegenerate affine hypersurface \(M\) with \(\dim M>2\) in the equiaffine space, the authors consider the isotropy condition for the difference tensor \(K\): NEWLINE\[NEWLINEh(K(X,X),K(X,X))=\lambda h(X,X)^2,NEWLINE\]NEWLINE where \(h\) is the affine metric of \(M\), \(X\) is any vector field on \(M\), and \(\lambda\) is a nonzero function on \(M\).NEWLINENEWLINEThe authors first show that the dimension of an isotropic affine hypersurface is either \(5\), \(8\), \(14\) or \(26\) by use of the Hurwitz \(1\), \(2\), \(4\), \(8\) theorem for composition of quadratic forms. Next, the authors obtain a complete classification of isotropic affine hyperspheres for each of the possible dimensions by determining the possible forms of the difference tensor.