Annals of Global Analysis and Geometry

, Volume 11, Issue 3, pp 197–211 | Cite as

RiemannianG-manifold with one-dimensional orbit space

  • Andrey V. Alekseevsky
  • Dmitry V. Alekseevsky


Cohomogeneity one RiemannianG-manifolds (i.e. Riemannian manifolds with a groupG of isometries having an orbit of codimension one) are studied. A description of such manifolds (up to some normal equivalence) is given in terms of Lie subgroups of Lie groupG. The twist of a geodesic normal to all orbits is defined as the number of intersections with a singular orbit. It is equal to the order of some Weyl group, associated with theG-manifold. Some results about possible values of the twist are obtained.

Key words

Riemannian G-manifold action of compact Lie group orbit space Weyl group normal geodesic sections cohomogeneous one manifold 

MSC 1991

53C35 53C30 53C21 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ale]
    Alekseevsky, D.V.: On a proper action of a Lie group.Uspekhi Mat. Nauk 34 (1979), 219–220.Google Scholar
  2. [Ale']
    Alekseevsky, D.V.: Riemannian manifolds of cohomogeneity one.Colloq. Math. Soc. J. Bolyai 56 (1989), 9–22.Google Scholar
  3. [Ale-Ale]
    Alekseevsky, A.V.;Alekseevsky, D.V.:G-manifold with one-dimensional orbit space.Adv. in Sov. Math. 8 (1992), 1–31.Google Scholar
  4. [Be-Ber]
    Bérard-Bergery, L.: Sur de nouvelles varietes riemanniennes d'Einstein.Publ. Inst. E. Cartan 4 (1982), 1–60.Google Scholar
  5. [Bess]
    Besse, A.L.:Einstein manifolds. Springer Verlag, 1987.Google Scholar
  6. [Bor]
    Borel, A.: Some remarks about Lie groups transitive on sphere and tori.Bull. Amer. Math. Soc. 55 (1949), 580–586.Google Scholar
  7. [Bott-Sam]
    Bott, R.;Samelson, H.: Applications of the theory of Morse to symmetric spaces.Amer. Math. Soc. 80 (1958), 964–1029.Google Scholar
  8. [Bred]
    Bredon, G.E.:Introduction to compact transformation groups. Acad. Press, N.Y. - London 1972.Google Scholar
  9. [Con]
    Conlon, L.: A class of variationally complete representations.J. Differential Geom. 7 (1972), 135–147.Google Scholar
  10. [Gro-Halp]
    Grove, K.;Halperin, S.: Duphin hypersurfaces, group actions and double mapping cylinder.J. Differential Geom. 26 (1987) 3, 429–460.Google Scholar
  11. [Most]
    Mostert, P.S.: On a compact Lie group acting on a manifold.Ann. of Math. 65 (1957), 447–455.Google Scholar
  12. [Pal]
    Palais, R.S.: On the existence of slices for actions of non-compact groups.Ann. of Math. 73 (1961), 295–323.Google Scholar
  13. [Pal-Ter]
    Palais, R.S.;Terng, Ch.L.: A general theory of canonical forms.Trans. Amer. Math. Soc. 300 (1987) 2, 771–789.Google Scholar
  14. [Sze]
    Szenthe, J.: Orthogonally transversal submanifolds and the generalization of the Weyl group.Period. Math. Hungar. 15 (1984), 281–299.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Andrey V. Alekseevsky
    • 1
  • Dmitry V. Alekseevsky
    • 2
  1. 1.Belozuorsky LaboratoryMoscow UniversityMoscowRussia
  2. 2.Center „Sophus Lie“MoscowRussia

Personalised recommendations