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On the laws of trigonometries of two-point homogeneous spaces. (English) Zbl 0695.53036
The paper deals with the trigonometry of triangles in symmetric spaces of rank 1. Apart from the cases of real type \(({\mathbb{E}}^ n,{\mathbb{S}}^ n,{\mathbb{H}}^ n)\), a triangle has nine basic invariants, namely the three side lengths and two angular invariants for any vertex. In the complex case for example, these two invariants for a pair of unit vectors are determined by \(\sphericalangle (u_ 1,u_ 2)\) and \(\sphericalangle (u_ 2,iu_ 1)\), and similarly for the quaternionic and Cayley type. Since, for the nonreal types, the moduli space of congruence classes of triangles has dimension 4, there must exist \(5=9-4\) trigonometric laws. The author gains these rules in a unified manner from a new, more general theorem which applies to abstract rotational manifolds where only the linear parts of the isometry group action are equivalent to the standard types.
Reviewer: R.Walter

MSC:
53C35 Differential geometry of symmetric spaces
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