Homogeneous submanifolds in the four-dimensional affine and projective geometry
The author describes all locally homogeneous hypersurfaces in the projective, resp. affine real 4-space with symmetry algebras of dimensions \(\geq 4\) by specifying 27, resp. 33 types of implicit equations. The general method follows \textit{B. Doubrov}, \textit{B. Komrakov} and \textit{M. Rabinovich} [Geom. and Top. of Submanifolds VIII, World Scientific, Singapore, 168-178 (1996; Zbl 0934.53007)]. Another classification of these surfaces is given by \textit{M. Eastwood} and \textit{V. Ezhov} [Tr. Mat. Inst. Steklova 235, 57-70 (2001; Zbl 1050.53041)] by a different method. Various further instances of the homogeneity problem are currently discussed in the literature; see e.g. the review for the paper of \textit{M. Eastwood} and \textit{V. Ezhov} in Geometriae Dedicata 77, No. 1, 11-69 (1999; Zbl 0999.53008). In particular, for the real equiaffine geometry, all homogeneous hypersurfaces in 4-space generated by 3-dimensional groups were determined by \textit{M. Wermann} [Diss. Univ. Dortmund 2001, Shaker Verlag (2001; Zbl 1028.53010)].
scientific article; zbMATH DE number 1758674