On locally symmetric affine hypersurfaces
The authors investigate 3-dimensional, locally strongly convex affine hyperspheres \(M\) in \(\mathbb{R}^ 4\) for which the Levi-Civita connection of the affine metric is locally symmetric. Their main result is a complete classification of these hypersurfaces which gives a complete classification of the 3-dimensional, locally strongly convex affine homogeneous hyperspheres, too. They obtain three types of hypersurfaces, where two are isometric with spaces of constant sectional curvature and the third is isometric with the locally symmetric space \(\mathbb{R} \times H^ 2\). For the proof they use special adapted frames which were introduced in earlier works of the second author. Finally they give a 4- dimensional example of a locally strongly convex affine homogeneous hypersphere for which the Levi-Civita connection of the affine metric is not locally symmetric.
scientific article; zbMATH DE number 687033