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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

Citations:

Zbl 0695.53037
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References:

[1] Arnold, V. I.: Mathematical Methods of Classical Mechanics, GTM, vol.60 (Springer-Verlag, 1980, New York- Heidelberg-Berlin). Arnold, V. I.: Mathematische Methoden der klassischen Mechanik (VEB Deutscher Verlag der Wissenschaften, 1988, Berlin, and Birkhäuser Verlag, 1988, Basel-Boston-Berlin).
[2] Bonola, R.: Non-Euclidean Geometry (Dover, 1955, New York).
[3] Borel, A.: Le plan projectif de octaves et les sphères comme espaces homogènes, Comptes Rendus Acad. Sci. Paris230 (1960), 1378-1380. · Zbl 0041.52203
[4] Cartan, È.: Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France54 (1926), 216-264 and55 (1927), 114-134. · JFM 53.0390.01
[5] Cartan, È.: La géométrie des groupes de transformations, J, Math. Pures Appl.6 (1927), 1-119. · JFM 53.0388.01
[6] Euclid: Elements. · Zbl 1026.01024
[7] Klein,F.: Vergleichende Betrachtungen über neuere geometrische Forschungen, Math. Ann.43 (1893), 63-100. · JFM 25.0871.01 · doi:10.1007/BF01446615
[8] Wang, H. C.: Two point homogeneous spaces, Ann. of Math.55 (1952), 177-191. · Zbl 0048.40503 · doi:10.2307/1969427
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