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Laws of trigonometry on $ {\rm SU}(3)$


Author: Helmer Aslaksen
Journal: Trans. Amer. Math. Soc. 317 (1990), 127-142
MSC: Primary 53C20; Secondary 15A72, 20G20, 53C35
MathSciNet review: 961593
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Abstract: The orbit space of congruence classes of triangles in $ SU(3)$ has dimension $ 8$. Each corner is given by a pair of tangent vectors $ (X,Y)$, and we consider the $ 8$ functions $ {\text{tr}}{X^2},i{\text{tr}}{X^3},{\text{tr}}{Y^2},i{\text{tr}}{Y^3},{\text{tr}}XY,i{\text{tr}}{X^2}Y,i{\text{tr}}X{Y^2}$ and $ {\text{tr}}{X^2}{Y^2}$ which are invariant under the full isometry group of $ SU(3)$. We show that these $ 8$ corner invariants determine the isometry class of the triangle. We give relations (laws of trigonometry) between the invariants at the different corners, enabling us to determine the invariants at the remaining corners, including the values of the remaining side and angles, if we know one set of corner invariants. The invariants that only depend on one tangent vector we will call side invariants, while those that depend on two tangent vectors will be called angular invariants. For each triangle we then have $ 6$ side invariants and $ 12$ angular invariants. Hence we need $ 18 - 8 = 10$ laws of trigonometry. If we restrict to $ SU(2)$, we get the cosine laws of spherical trigonometry. The basic tool for deriving these laws is a formula expressing $ {\text{tr}}({\operatorname{exp}}X{\operatorname{exp}}Y)$ in terms of the corner invariants.


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  • [BT] Wilhelm Blaschke and Hans Terheggen, Trigonometria hermitiana, Rend. Sem. Mat. Roma 3 (1939), 153–161 (Italian). MR 0001572
  • [Br] U. Brehm, The shape invariant of triangles in two-point homogeneous spaces, unpublished manuscript.
  • [Co] J. L. Coolidge, Hermitian metrics, Ann. of Math. (2) 22 (1920), no. 1, 11–28. MR 1502568, 10.2307/1967718
  • [Du1] Ja. S. Dubnov, Sur une generalisation de l'équation de Hamilton-Cayley et sur les invariants simultanes de plusieurs affineurs, Trudy. Sem. Vektor. Tenzor. Anal. 2-3 (1935), 351-367.
  • [Du2] Ya Dubnov, Complete system of invariants of two affinors in centro-affine space of two or three dimensions, Abh. Sem. Vektor- und Tensoranalysis [Trudy Sem. Vektor. Tenzor. Analizu] 5 (1941), 250–270 (Russian). MR 0016983
  • [Hs] Wu-Yi Hsiang, On the laws of trigonometries of two-point homogeneous spaces, Ann. Global Anal. Geom. 7 (1989), no. 1, 29–45. MR 1029843, 10.1007/BF00137400
  • [Le1] John S. Lew, The generalized Cayley-Hamilton theorem in 𝑛 dimensions, Z. Angew. Math. Phys. 17 (1966), 650–653 (English, with French summary). MR 0213381
  • [Le2] John S. Lew, Reducibility of matrix polynomials and their traces, Z. Angew. Math. Phys. 18 (1967), 289–293 (English, with French summary). MR 0210725
  • [Lo] O. Loos, Symmetric spaces, Benjamin, 1969.
  • [Ri] R. S. Rivlin, Further remarks on the stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4 (1955), 681–702. MR 0071980
  • [Ro] B. A. Rozenfel′d, On the theory of symmetric spaces of rank one, Mat. Sb. N.S. 41(83) (1957), 373–380 (Russian). MR 0096271
  • [Sib] K. S. Sibirskiĭ, Algebraic invariants of a system of matrices, Sibirsk. Mat. Z. 9 (1968), 152–164 (Russian). MR 0223379
  • [Sir] P. A. Širokov, On a certain type of symmetric spaces, Mat. Sb. N.S. 41(83) (1957), 361–372 (Russian). MR 0096269
  • [Sp] A. Cemal Eringen (ed.), Continuum physics. Vol. I, Academic Press, New York-London, 1971. Mathematics. MR 0468443
  • [Te] H. Terheggen, Zur analytischen Geometrie auf der Geraden von Hermite als Grenzfall der Geometrie in der Hermitischen Ebene und ihr Zusammenhang mit der gewöhnlichen sphärischen Trigonometrie, Jber. Deutsch. Math. Verein. 50 (1940), 24–35 (German). MR 0003030
  • [Wa] Hsien-Chung Wang, Two-point homogeneous spaces, Ann. of Math. (2) 55 (1952), 177–191. MR 0047345

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0961593-7
Keywords: Invariants, trigonometry, Lie groups, symmetric spaces
Article copyright: © Copyright 1990 American Mathematical Society