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On the space problem of Helmholtz. (English) Zbl 0944.20029
The first part of the paper is a brief history of Helmholtz’s space problem. In the second part the author proves characterizations of a closed subgroup $$K$$ of $$\text{GL}(n,R)$$ to be the orthogonal group of a certain positive definite real symmetric matrix of degree $$n$$: (i) $$K$$ is equivalent to $$O(n)$$ by an element of the Iwasawa subgroup $$T$$ of $$\text{GL}(n,R)$$; or (ii) $$\text{GL}(n,R)$$ is the direct product of $$K$$ with $$T$$; or (iii) $$K$$ is a maximal compact subgroup; or (iv) $$K$$ acts simply transitively on the flag-manifold of $$R^n$$, this the author calls the condition of free mobility.
##### MSC:
 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 14L35 Classical groups (algebro-geometric aspects) 51M05 Euclidean geometries (general) and generalizations 51N30 Geometry of classical groups