The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces
The authors give a complete classification of the 3-dimensional homogeneous, locally strongly convex affine hypersurfaces of \(\mathbb{R}^ 4\), according to the number of different eigenvalues of their shape operator \(S\). So they prove that there are no 3-dimensional homogeneous, locally strongly convex affine hypersurfaces of \(\mathbb{R}^ 4\), whose shape operator \(S\) has three distinct eigenvalues. Next, considering that \(S\) has two distinct eigenvalues and being based on their previous papers they point out that a 3-dimensional homogeneous, locally strongly convex affine hypersurface \(M\) of \(\mathbb{R}^ 4\) is affine equivalent to the convex part of one of four explicit given hypersurfaces. Finally, if \(S\) is a multiple of the identity, \(M\) is a 3-dimensional locally strongly convex, locally homogeneous hypersphere in \(\mathbb{R}^ 4\) and then it is shown that either \(M\) is an open part of a locally strongly convex quadric or \(M\) is affine equivalent to an open part of one of the hypersurfaces \(xyzw=1\), \((y^ 2- z^ 2- w^ 2)^ 3x^ 2=1\).
scientific article; zbMATH DE number 492798