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On affine normal forms and a classification of homogeneous surfaces in affine three-space. (English) Zbl 0999.53008
The authors contribute to the homogeneity problem of affine hypersurfaces by establishing a new method based on the Taylor expansion for the defining graph function with respect to the Blaschke/Berwald normal as last axis. Specifically, they classify all the affinely homogeneous surfaces in \(3\)-space, thus confirming the result of B. Doubrov, B. Komrakov and M. Rabinovich [Geometry and Topology of Submanifolds VIII, World Scientific, Singapore, 168-178 (1996; Zbl 0934.53007)]. The main difference is that instaed of normalizing the symmetry algebras now the graph functions are normalized. Actually, this is done by a clever induction procedure on the Taylor polynomials of these functions. The authors believe that their approach is more advantegeous. As they realized after the completion of their work, yet another method for the classification of these surfaces has been proposed by A. V. Loboda [Math. Notes 65, No. 5, 668-672 (1999); translation from Mat. Zametki 65, 793-797 (1999; Zbl 0976.53012) and Proc. ‘Pontrjagin’s Readings-6’, Voronezh-98, 124 (1998)].
Several computations related to the problem of homogeneity are nonlinear and rather complex, including enormous branching phenomena. So machine computing is sometimes inevitable. On the other hand, there doesn’t exist an automatism to attack the problem as a whole. For the next higher dimension (hypersurfaces in \(4\)-space) there are recent works by the same authors in the complex domain [Asian J. Math. 5, 721-740 (2001; Zbl 1022.53043)] and by M. Wermann with respect to real classification [Dissertation Univ. Dortmund 2001, Shaker Verlag (2001; Zbl 1028.53010)]. Codimension two homogeneity problems in \(4\)-space have been solved by normalizing Lie subalgebras in the reviewer’s papers [Geom. Dedicata 71, 129-178 (1998; Zbl 0973.53511), Contrib. Algebra Geom. 41, No. 1, 159-180 (2000; Zbl 0976.53013)], and forthcoming [Banach Center Publications, Proc. Conf. ‘PDEs, submanifolds and affine differential geometry’, Banach Center, Warzaw, Sept. 2000].

MSC:
53A15 Affine differential geometry
53C30 Differential geometry of homogeneous manifolds
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