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On a theorem of Frobenius in the \(n\)-dimensional isotropic space \(I^k_n\). (Über einen Satz von Frobenius im \(n\)-dimensionalen isotropen Raum \(I^k_n\).) (German) Zbl 1157.51014

Denote by \(H\) the subgroup of the isotropic affine group, described by block-diagonal matrices with two diagonal elements: An orthogonal matrix \(\mathbf A\) of dimension \((n-k,n-k)\) and a regular matrix \(\mathbf C\) of dimension \((k,k)\).
The authors prove the following result: If \(G\) is a finite subgroup of the affine group in isotropic spaces such that there exists an isotropic vector \(\mathbf x\) whose orbit \(\{\lambda({\mathbf x})\mid\lambda\in G\}\) has maximal Euclidean dimension then \(G\) is conjugate to a finite subgroup of \(H\).
For \(k = 0\) this is just the Theorem of Frobenius for the \(n\)-dimensional Euclidean space. In contrast to the Euclidean case, the conjugate subgroup \(\alpha\circ G\circ\alpha^{-1}\) is not necessarily contained in the isotropic motion group (the subgroup where \(\mathbf C\) is a lower triangular matrix).
The notions and definitions used in the reviewed article (and in this review) can be found in H. Vogler and H. Wresnik [Grazer Math. Ber. 307, 46 p. (1989; Zbl 0682.53012)].

MSC:

51N25 Analytic geometry with other transformation groups
53A17 Differential geometric aspects in kinematics
53A35 Non-Euclidean differential geometry

Citations:

Zbl 0682.53012
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