Laws of trigonometry in symmetric spaces

This paper consists of two parts. In the first part, we reformulate the work of E. Leuzinger on trigonometry in noncom pact 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 comparison theorem for geodesic triangles in the form of an inequality. On the other hand, it is clear that congruence conditions among configurations of points can only be found in spaces with a high degree of symmetry. Only in such spaces can we obtain exact formulas relating invariant geometric quantities. We will hence consider a globally symmetric space X. We will also be interested in configurations of n points, where n may be greater than 3, and we will refer to this as trigonometry, too. Let Xli be the space of ordered subsets of n points. Given a group of isometries G of X, we let G act diagonally on Xli. That is, (x I, . ,Xli) -+ (gxl, . ,gxlI ) for g E G. Let U be a G-invariant open dense subset of Xli. Once U is specified, an n-tuple (XI, . ,Xli) E U is said to be in general position. A trigonometric quantity with respect to G is a G-invariant function a: U -+ F, where F is a set on which G acts trivially. We will also call the induced function Of/print from: Geometry from the Pacific Rim, Eds.: Berrick/LoofWang © Walter de Gruyter & Co., Berlin' New York 1997 24 H. Aslaksen and H. -L. Huynh a: V / G ~ F a trigonometric quantity. Let R be a subset of F. We say that a satisfies the trigonometric relation R if a(V) S; R. Finally, a is said to be a congruence condition with respect to G if it is an injective function into F. For example, let X be the Euclidean plane and let V C X 3 consist of the 3-tuples of distinct points. Then a: V ~ lR, X lR, x lR, X 51, which maps an ordered triangle to its three sides and the angle subtended by the first two sides, is a trigonometric quantity with respect to the group of orientation-preserving isometries. It satisfies the trigonometric relation 2 {(a, b, c,8) I a +b c 2abcos8 = OJ. It is also a congruence condition with respect to the same group. 2. The moduli space of n points in symmetric spaces By definition, a Riemannian (globally) symmetric space X is a Riemannian manifold with central symmetry through every point x E X, i.e., the diffeomorphism that reverses all the geodesics through x is a global isometry. We will make use of some basic facts about symmetric spaces. Proofs and details can be found in [He], [Mo] or [Wo]. Let I (X) be the full isometry group of X, and let 10(X) be the component of the identity. Then 10(X) acts transitively on X. Let G be a Lie group such that 10(X) S; G S; I(X) and pick a basepoint xo E X. Let K be the subgroup of G that fixes xo and let a be the geodesic symmetry at xo. Then X = G / K, and (Ga)O S; K S; G a, where G a is the set of fixed points of a and (Ga)O is the identity component of Ga. If X is of noncom pact type, then K will be a maximal compact subgroup of G. We would like to determine X n / G or V / G, where V is a suitably chosen set of points in general position. Let p be the tangent space of X atxo. Underthe exponential mapExp: p ~ X, we have a correspondence X n /G ++ (Eft p)/ K in the noncompact case, and V /G ++ (Elt p)/K in the compact case. The ring of invariant polynomial functions, lR[Ett p]K, is the coordinate ring of the space (EB p)/K, and we wish to describe the moduli space of n points by giving a set of generators and relations for this ring. Note that the generators will then be trigonometric quantities, and the relations will be trigonometric relations. For the classical geometries, the full isotropy group is Oem). The isotropy rep­ resentation is the standard action on lR,m, and the generators and relations of the ring lR[EB p]K are given by the First and Second Fundamental Theorems of invariant the­ ory of H. Weyl ([We]). For symmetric spaces of rank greater than one, our knowledge of lR[EB p]K is less complete. On the other hand, the geometry of two points in a symmetric space is well un­ derstood. Consider the set of flats in X. These are the maximal, flat, totally geodesic subspaces. They are all of the same dimension r, which is the rank of the symmetric 25 Laws of trigonometry in symmetric spaces space X. Each flat is isometric to a Euclidean space in case X is of noncompact type, and a flat torus in case X is compact. Every point of X is contained in a flat, G per­ mutes the flats transitively and K permutes transitively all those that contain the point Xa. Given a flat F, there is an Abelian subgroup A of G that restricts to the full group of Euclidean translations on F. Let us choose a base flat Fa containing xa. Then each x E Fa is axa for some a E A, and each x E X is ky for some k E K and y E Fa. It follows that if we have two points in X, then we may assume that the first point is xa and the second point is x I E Fa. Hence X2 / G ~ Fa/ M', where M' is the subgroup of K that preserves the set Fa. Under the correspondence a -+ axa, the action of M' on Fa is equivalent to the action of M' on A given by a -+ maml , and M' is the normalizer of A in K. Furthermore, M = ZK(A), the centralizer of A in K, is the subgroup of K that fixes Fa pointwise. The Weyl group W = M' / M decomposes Fa into a union of Weyl chambers. For each chamber C, every W -orbit in Fa intersect C exactly once. And W permutes the set of chambers simply and transitively. Now let us choose among the chambers a base chamber Ca, so that it contains the point Xl. Collecting all these facts together, we obtain the following description of the moduli of 2 points. X 2 /G ~ K\G/K ~ A/W ~Ca. To specify a pair of points (y, y') up to congruence with respect to the full group of isometries I (X), we first move them by an element of g E G to a pair (g y, gy') = (xa, Xl) = (xa, axa), where xa is the base point, XI lies in the base chamber and a E A. Define M* s;: I (X) to be the subgroup that fixes the point xa and preserves the set Ca. Then The coset M*a is called the interval determined by the ordered pair of points (y, y'). The structure of lR[p]K is completely determined by a theorem of Chevalley. Let 21 be the linear subspace of p that exponentiates to Fa. The adjoint action of K on p restricts to an action of M' on 21, with kernel M. Hence the group W acts on 21, and we can state the following theorem.