## 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

### Keywords:

isotropic geometry; Theorem of Frobenius

Zbl 0682.53012