Classification of flat indefinite equicentroaffinely homogeneous surfaces in \(\mathbf R^ 4\)
Let \(x: M\to \mathbb{R}^4\) be an immersion of a surface. For any local basis \(\sigma=\{E_1,E_2\}\) one can locally define a symmetric 2-form \(G^\sigma=[E_1(x),E_2(x),x,d^2x] =\Sigma G^\sigma_{ij} \theta_i \theta_j\), where \(\{\theta_1,\theta_2\}\) is the dual basis for \(\sigma\). \(x\) is called nondegenerate if \(G^\sigma\) is a nondegenerate 2-form. It is known that for any nondegenerate surface \(x\) one gets a globally defined equi-centroaffinely invariant metric \(G:=|\text{det}G^\sigma_{ij}|^{1/4}G^\sigma\). The author uses \(G\) to define a complete system of equi-centroaffine invariants \(\{\Lambda,\Phi,\Psi,\Theta\}\) (where \(\Phi\) and \(\Lambda\) are 3-forms, \(\Psi\) and \(\Theta\) are 2-forms), which determines the surface \(x\) up to equi-centroaffine transformations in \(\mathbb{R}^4\). Using this invariant system, the author is able to determine all the nondegenerate flat equi-centroaffine homogeneous surfaces in \(\mathbb{R}^4\) which are not contained in any hyperplane of \(\mathbb{R}^4\). There are 16 types of such homogeneous surfaces. The author also gives examples of equi-centroaffine homogeneous surfaces which are orbits of two different centroaffine subgroups.
scientific article; zbMATH DE number 920812