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

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

MathSciNet review:
773051

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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]**J. Dadok and V. Kac,*Polar representations*, J. Algebra (to appear). MR**778464 (86e:14023)****[2]**J. Dadok and F. R. Harvey,*Calibrations in*, Duke Math. J.**50**(1983), 1231-1243. MR**726326 (85a:53056)****[3]**S. Helgason,*Differential geometry and symmetric spaces*, Academic Press, New York, 1981. MR**0145455 (26:2986)****[4]**-,*Analysis on Lie groups and homogeneous spaces*, CBMS Regional Conf. Ser. in Math., no. 14, Amer. Math. Soc., Providence, R. I., 1972. MR**0316632 (47:5179)****[5]**J. E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York, 1972. MR**0323842 (48:2197)****[6]**G. A. Hunt,*A theorem of Élie Cartan*, Proc. Amer. Math. Soc.**7**(1956), 307-308. MR**0077075 (17:986c)**

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Additional Information

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

Keywords:
Orbits,
linear cross sections

Article copyright:
© Copyright 1985
American Mathematical Society