Summary: Laws of trigonometry in symmetric spaces
Helmer Aslaksen and Hsueh-Ling Huynh
Abstract. This paper consists of two parts. In the first part, we reformulate the work of
E. Leuzinger on trigonometry in noncompact symmetric spaces. In the second part, we outline
an altemative method using invariants of the isotropy group representation. Appropriately
formulated, these methods apply to both compact and noncompact symmetric spaces. This
work is contained in the Ph.D. dissertation of H.-L. Huynh.
1991 Mathematics Subject Classification: 53C35, 15A72, 15A24
1. What is trigonometry?
Trigonometry is fundamental to the study of the classical geometries. First of all,
it gives congruence conditions for triangles. In Euclidean, hyperbolic and spherical
spaces, two triangles are congruent if and only if they satisfy the side-angle-side (SAS)
or side-side-side (SSS) condition.
Given a triangle, we also want to associate to it metric quantities that are invariant
under the group of isometries. The laws of trigonometry can then be interpreted as
relations between such quantities. Given two sides and their subtended angle, the law
of cosines determines the third side, and given two sides and one angle adjacent to the
third side, the law of sines determines the second adjacent angle.
In a general Riemannian manifold, the law of cosines can be generalized as a