Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [BT] W. Blaschke and H. Terheggen, Trigonometria Hermitiana, Rend. Sem. Mat. Univ. Roma (4) 3 (1939), 153-161. MR 0001572 (1:261d)
  • [Br] U. Brehm, The shape invariant of triangles in two-point homogeneous spaces, unpublished manuscript.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 (1921), 11-28. MR 1502568
  • [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] -, Complete system of invariants of two affinors in centro-affine space of two or three dimensions, Trudy. Sem. Vektor. Tenzor. Anal. 5 (1941), 250-270. (Russian) MR 0016983 (8:95a)
  • [Hs] W.-Y. Hsiang, On the trigonometry of two-point homogeneous space (Preprint, Berkeley, 1986), Ann. Global Anal. Geom. (to appear). MR 1029843 (90k:53086)
  • [Le1] J. S. Lew, The generalized Cayley-Hamilton theorem in $ n$ dimensions, Z. Angew. Math. Phys. 17 (1966), 650-653. MR 0213381 (35:4245)
  • [Le2] -, Reducibility of matrix polynomials and their traces, Z. Angew. Math. Phys. 18 (1967), 289-293. MR 0210725 (35:1611)
  • [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 (17:210a)
  • [Ro] B. A. Rozenfeld, On the theory of symmetric spaces of rank one, Mat. Sb. (N.S.) 41(83) (1957), 373-380. (Russian) MR 0096271 (20:2756a)
  • [Sib] K. S. Sibirskii, Algebraic invariants for a set of matrices, Siberian Math. J. 9 (1968), 115-124. MR 0223379 (36:6427)
  • [Sir] P. A. Sirokov, On a certain type of symmetric space, Mat. Sb. (N.S.) 41(83) (1957), 361-372. (Russian) MR 0096269 (20:2755a)
  • [Sp] A. J. M. Spencer, The theory of invariants, Continuum Physics (A. C. Eringen, ed.), Academic Press, 1971, pp. 239-353. MR 0468443 (57:8277a)
  • [Te] H. Terheggen, Zur Analytischen Geometrie auf der Geraden von Hermite als Grenzfall der Geometrie in der Hermitischen Ebene and ihr Zusammenhang mit der gewàhnlichen sphärischen Trigonometrie, Jahresber. Deutsch. Math.-Verein. 50 (1940), 24-35. MR 0003030 (2:152c)
  • [Wa] H. C. Wang, Two-point homogeneous spaces, Ann. of Math. (2) 55 (1952), 172-191. MR 0047345 (13:863a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20, 15A72, 20G20, 53C35

Retrieve articles in all journals with MSC: 53C20, 15A72, 20G20, 53C35

Additional Information

Keywords: Invariants, trigonometry, Lie groups, symmetric spaces
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society