zbMATH — the first resource for mathematics

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.
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