Polar coordinates induced by actions of compact Lie groups

Author:
Jiri Dadok

Journal:
Trans. Amer. Math. Soc. **288** (1985), 125-137

MSC:
Primary 22E15; Secondary 53C35

MathSciNet review:
773051

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a connected Lie subgroup of the real orthogonal group . For the action of on , we construct linear subspaces that intersect all orbits. We determine for which there exists such an meeting all the -orbits orthogonally; groups that act transitively on spheres are obvious examples. With few exceptions all possible arise as the isotropy subgroups of Riemannian symmetric spaces.

**[1]**Jiri Dadok and Victor Kac,*Polar representations*, J. Algebra**92**(1985), no. 2, 504–524. MR**778464**, 10.1016/0021-8693(85)90136-X**[2]**Jiri Dadok and Reese Harvey,*Calibrations on 𝑅⁶*, Duke Math. J.**50**(1983), no. 4, 1231–1243. MR**726326**, 10.1215/S0012-7094-83-05053-6**[3]**Sigurđur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[4]**Sigurdur Helgason,*Analysis on Lie groups and homogeneous spaces*, American Mathematical Society, Providence, R.I., 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 14. MR**0316632****[5]**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842****[6]**G. A. Hunt,*A theorem of Elie Cartan*, Proc. Amer. Math. Soc.**7**(1956), 307–308. MR**0077075**, 10.1090/S0002-9939-1956-0077075-9

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22E15,
53C35

Retrieve articles in all journals with MSC: 22E15, 53C35

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1985-0773051-1

Keywords:
Orbits,
linear cross sections

Article copyright:
© Copyright 1985
American Mathematical Society