Indefinite equi-centroaffinely homogeneous surfaces with vanishing Pick-invariant in \(\mathbb{R}^ 4\)
The author determines all surfaces in \( \mathbb{R}^4 \) which are homogeneous with respect to the special linear group \(\text{SL} (4,\mathbb{R})\) with indefinite equi-centroaffine metric under the additional assumption of vanishing Pick invariant. Without loss of generality, surfaces contained in an affine hyperplane can be excluded because they are affinely homogeneous in \( \mathbb{R}^3 \). There result 15 cases which are described by explicit parametrizations. In previous papers the author has similarly treated homogeneous surfaces under the side conditions \(K=0\) [\textit{H.-L. Liu}, Result. Math. 29, 280-304 (1996; Zbl 0860.53009)]), resp. \(J=0\) for definite metric [Proc. 1-st Non-Orthodox School on Nonlinearity and Geometry, Warszawa 21-28 September 1995 (Banach Center Publ., Warsaw) (to appear)]. See, also for the 3-dimensional situation, the author's thesis [`Zentroaffine homogene Flächen im \(\mathbb{R}^3\) und \(\mathbb{R}^4\)', TU Berlin (1996)].
scientific article; zbMATH DE number 992424