Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 093, 16 pages      arXiv:0904.3592      doi:10.3842/SIGMA.2009.093
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Compact Riemannian Manifolds with Homogeneous Geodesics

Dmitrii V. Alekseevsky a and Yurii G. Nikonorov b
a) School of Mathematics and Maxwell Institute for Mathematical Studies, Edinburgh University, Edinburgh EH9 3JZ, United Kingdom
b) Volgodonsk Institute of Service (branch) of South Russian State University of Economics and Service, 16 Mira Ave., Volgodonsk, Rostov region, 347386, Russia

Received April 22, 2009, in final form September 20, 2009; Published online September 30, 2009

Abstract
A homogeneous Riemannian space (M = G/H,g) is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group G. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric g with homogeneous geodesics on a homogeneous space of a compact Lie group G. We give a classification of compact simply connected GO-spaces (M = G/H,g) of positive Euler characteristic. If the group G is simple and the metric g does not come from a bi-invariant metric of G, then M is one of the flag manifolds M1 = SO(2n+1)/U(n) or M2 = Sp(n)/U(1)·Sp(n–1) and g is any invariant metric on M which depends on two real parameters. In both cases, there exists unique (up to a scaling) symmetric metric g0 such that (M,g0) is the symmetric space M = SO(2n+2)/U(n+1) or, respectively, CP2n–1. The manifolds M1, M2 are weakly symmetric spaces.

Key words: homogeneous spaces, weakly symmetric spaces, homogeneous spaces of positive Euler characteristic, geodesic orbit spaces, normal homogeneous Riemannian manifolds, geodesics.

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References

  1. Alekseevsky D.V., Arvanitoyeorgos A., Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc. 359 (2007), 3769-3789.
  2. Akhiezer D.N., Vinberg E.B., Weakly symmetric spaces and spherical varieties, Transform. Groups 4 (1999), 3-24.
  3. Berestovskii V.N., Nikonorov Yu.G., On δ-homogeneous Riemannian manifolds, Differential Geom. Appl. 26 (2008), 514-535, math.DG/0611557.
  4. Berestovskii V.N., Nikonorov Yu.G., On δ-homogeneous Riemannian manifolds. II, Siber. Math. J. 50 (2009), 214-222.
  5. Berestovskii V.N., Nikitenko E.V., Nikonorov Yu.G., The classification of δ-homogeneous Riemannian manifolds with positive Euler characteristic, arXiv:0903:0457.
  6. Berger M., Les variétés riemanniennes homogènes normales à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 179-246.
  7. Berndt J., Kowalski O., Vanhecke L., Geodesics in weakly symmetric spaces, Ann. Global Anal. Geom. 15 (1997), 153-156.
  8. Besse A.L., Einstein manifolds, Springer-Verlag, Berlin, 1987.
  9. Borel A., de Siebenthal J., Les sous-groups fermés de rang maximum des groups de Lie clos, Comment. Math. Helv. 23 (1949), 200-221.
  10. Cartan É., Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France 54 (1926), 214-264.
    Cartan É., Sur une classe remarquable d'espaces de Riemann. II, Bull. Soc. Math. France 55 (1927), 114-134.
  11. D'Atri J.E., Ziller W., Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), no. 215, 1-72.
  12. Dusek Z., Kowalski O., Nikcevic S., New examples of Riemannian g.o. manifolds in dimension 7, Differential Geom. Appl. 21 (2004), 65-78.
  13. Gorbatzevich V.V., Onishchik A.L., Vinberg E.B., Lie groups and Lie algebras. III. Structure of Lie groups and Lie algebras, Encyclopaedia of Mathematical Sciences, Vol. 41, Springer-Verlag, Berlin, 1994.
  14. Gordon C.S., Homogeneous Riemannian manifolds whose geodesics are orbits, in Topics in Geometry: in Memory of Joseph D'Atri, Progr. Nonlinear Differential Equations Appl., Vol. 20, Birkhäuser Boston, Boston, MA, 1996, 155-174.
  15. Helgason S., Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. 12, Academic Press, New York - London, 1962.
  16. Kaplan A., On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1990), 32-42.
  17. Kerr M.M., Some new homogeneous Einstein metrics on symmetric spaces, Trans. Amer. Math. Soc. 348 (1996), 153-171.
  18. Kobayashi S., Nomizu K., Foundations of differential geometry, John Wiley & Sons, New York, Vol. I, 1963, Vol. II, 1969.
  19. Kostant B., On holonomy and homogeneous spaces, Nagoya Math. J. 12 (1957), 31-54.
  20. Kowalski O., Vanhecke L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B (7) 5 (1991), 189-246.
  21. Onishchik A.L., Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994.
  22. Selberg A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces, with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87.
  23. Tamaru H., Riemannin G.O. spaces fibered over irreducible symmetric spaces, Osaka J. Math. 15 (1998), 835-851.
  24. Tamaru H., Riemannin geodesic orbit space metrics on fiber bundles, Algebra Groups Geom. 36 (1999) 835-851.
  25. Wolf J.A., Spaces of constant curvature, Publish or Perish, Inc., Houston, TX, 1984.
  26. Wolf J.A., Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, Vol. 142, American Mathematical Society, Providence, RI, 2007.
  27. Ziller W., Weakly symmetric spaces, in Topics in Geometry: in Memory of Joseph D'Atri, Progr. Nonlinear Differential Equations Appl., Vol. 20, Birkhäuser Boston, Boston, MA, 1996, 355-368.
  28. Yakimova O.S., Weakly symmetric Riemannian manifolds with a reductive isometry group, Sb. Math. 195 (2004), 599-614.

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