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Highly Symmetric Homogeneous Spaces

Published online by Cambridge University Press:  20 November 2018

L. N. Mann
L. N. Mann*
Affiliation:
University of Massachusetts, Amherst Massachusetts
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We consider effective homogeneous spaces M = G/H where G is a compact connected Lie group, H is a closed subgroup and G acts effectively on M (i.e., H contains no non-trivial subgroup normal in G). It is known that dim G ≦ m2/2 + m/2 where m = dim M and that if dim G = m2/2 + m/2, then M is diffeomorphic to the standard sphere Sm or the standard real projective space RPm [1]. In addition it has been shown that for fixed m there are gaps in the possible dimensions for G below the maximal bound [4; 5].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 2 , April 1974 , pp. 291 - 293
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Birkhoff, Garret, Extensions of Lie groups, Math. Z. 53 (1950), 226–235.Google Scholar
2. Borel, A., Transformation groups with two classes of orbits, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 983–985.Google Scholar
3. Ku, H. T., Mann, L. N., Sicks, J. L., and Su, J. C., Degree of symmetry of a product manifold, Trans. Amer. Math. Soc. 146 (1969), 133–149.Google Scholar
4. Mann, L. N., Gaps in the dimensions of transformation groups, Illinois J. Math. 10 (1966), 532–546.Google Scholar
5. Mann, L. N., Further gaps in the dimensions of transformation groups, Illinois J. Math. 15 (1969), 740–756.Google Scholar