Classification of homogeneous submanifolds in homogeneous spaces
This is a survey paper about homogeneous submanifolds in homogeneous spaces \(G/G_0\). The results are treated by a unified method based on the language of Lie algebras. There is given an explicit description of the stabilizers \(G_k\) of \(k\)-jets of a homogeneous submanifold, which form a decreasing chain of subgroups in the Lie group \(G\). It is proved that this chain always stabilizes. (This result has previously been obtained in [the reviewer, Ann. Math. Soc., Transl. II. Ser. 92, 231-274 (1970; Zbl 0206.50601)] by means of Maurer-Cartan forms.) Some auxiliary results lead to the fact that a locally homogeneous submanifold is always an open part of a globally homogeneous submanifold. All this allows to reduce the problem of classification of the latter to the purely algebraic problem of classifying subalgebras in Lie algebras. This method is illustrated by the classification of homogeneous curves in 2-dimensional real affine or projective geometry. Also, the results are summarized about classification of homogeneous surfaces in 3-dimensional real affine or projective geometry. They are given here without proofs, which can be found in preprints and papers (to appear) by the authors, with some coauthors (F. Dillen, M. Rabinovich). The same is done with homogeneous hypersurfaces in \(A^4\) and \(\mathbb{R} P^4\), where the recent results of \textit{N. Mozhei} [Izv. Vyssh. Uchebn. Zaved., Mat. 2000, No. 7, 41-52 (2000)] are summarized. All this is represented briefly in tables where all homogeneous surfaces or hypersurfaces are listed together with their equations in suitable coordinates.
scientific article; zbMATH DE number 1413886