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The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both lengthwise and angular) between pairs of lines. There is also a converse statement. Suppose we are living in a complete Riemannian manifold of constant curvature. If a certain sine/cosine rule is satsifies by triples of lines, then we can determine the curvature.

I'm wondering if we can generalize these sine and cosine rules to arbitrary symmetric spaces. That is, give a triple of geodesics (or parallel submanifolds, if we consider higher dimensions), are there similar invariants? Perhaps these invariants will be given in terms of representations of the coset of symmetries take a geodesic to antoher.

It would also be great if these invariants can characterize the symmetric space we are living in, just like in the case of the constant curvature case.

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up vote 6 down vote accepted

There exist generalizations of the trigonometry laws to symmetric spaces. The following article by Ortega and Santander works out the trigonometry laws for the case of real symmetric spaces of constant curvature and the following one the case of rank-1 Hermitian symmetric spaces. The following article by Aslaksen and Huynh treats more general cases of n-point trigonometrey of symmetric spaces.

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