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Homogeneous hypersurfaces in four-dimensional equiaffine space. (Homogene Hyperflächen im vierdimensionalen äqui-affinen Raum.) (German) Zbl 1028.53010

Berichte aus der Mathematik. Aachen: Shaker Verlag. Dortmund: Univ. Dortmund (Diss.), 229 S. (2001).
This thesis gives a complete classification of equiaffine homogeneous hypersurfaces in \(\mathbb R^4\). In this context equiaffine homogeneous immersions are immersions generated by an \(n\)-dimensional subgroup of the special affine group \(SA(n+1)\). Moreover, the immersions have to be regular which is equivalent to the (global) non-degeneracy of their (Euclidean) second fundamental form.
The classification of homogeneous equiaffine surfaces in \(\mathbb R^3\) has been completed in [B. Doubrov, B. Komrakov and M. Rabinovich, Dillen, F. (ed.) et al., Geometry and Topology of Submanifolds, VIII. Singapore: World Scientific, 168-178 (1996; Zbl 0934.53007)]. The classification of homogeneous hypersurfaces in \(\mathbb R^4\) is quite more complicated and has been studied up to now with additional assumptions on the immersion (see e.g. [F. Dillen and L. Vrancken, Manuscr. Math. 80, 165-180 (1993; Zbl 0795.53008)] and others).
In brevity the results are obtained in the following way: The classification is based on the classification of \(3\)-dimensional Lie algebras given in [N. Jacobson, Lie Algebras, Interscience Publisher (1962; Zbl 0121.27504)]. First, the given classes, characterized by relations of their generators, are tested – using the Jordan normal form of the generators – to figure out the subclasses which are able to generate regular hypersurfaces. Finally the dependence of the initial point of the orbit is discussed. This results in about \(40\) group classes and about \(60\) orbit families.
Then, the higher order contact is used to separate the classes of generated immersions. Whereas quadrics and generalized translational surfaces are exceptional cases, the remaining immersions determine their generating groups uniquely. Since there are many cases to consider and the separation by contact is computational quite extensive the author carefully used computer algebra programs to tackle this problem. A list of implicitly described immersions is given at the end of the thesis.

MSC:

53A15 Affine differential geometry
53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
68W30 Symbolic computation and algebraic computation
22Fxx Noncompact transformation groups
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