Annals of Global Analysis and Geometry

, Volume 55, Issue 3, pp 451–477 | Cite as

Variational aspects of homogeneous geodesics on generalized flag manifolds and applications

  • Rafaela F. do Prado
  • Lino GramaEmail author


We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space \({\mathbb {C}}P^{2n+1} = \text {Sp}(n+1)/(\text {U}(1)\times \text {Sp}(n))\).


Geometry of homogeneous space Homogeneous geodesics Conjugate points Morse index Generalized flag manifolds 


  1. 1.
    Abiev, N.A., Nikonorov, Y.G.: The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow. Ann. Glob. Anal. Geom. 50(1), 65–84 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekseevsky, D., Arvanitoyeorgos, A.: Riemannian flag manifolds with homogeneous geodesics. Trans. Am. Math. Soc. 359, 3769–3789 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bohm, C., Wilking, B.: Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. GAFA Geom. Funct. Anal. 17, 665–681 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burstall, F., Rawnsley, J.: Twistor theory for Riemannian symmetric spaces. With applications to harmonic maps of Riemann surfaces. Lecture Notes in Mathematics, vol. 1424. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chavel, I.: Isotropic Jacobi fields and Jacobi’s equations on Riemannian homogeneous spaces. Comment. Math. Helv. 42, 237–248 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheeger, J., Ebin, D.: Comparison Theorems in Riemannian Geometry. AMS Chelsea Publishing, Providence (1975)zbMATHGoogle Scholar
  7. 7.
    Cohen, N., Grama, L., Negreiros, C.J.C.: Equigeodesics on flag manifolds. Houst. Math. J. 37(1), 113–125 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    González-Dávila, J.C., Naveira, A.M.: Existence of non-isotropic conjugate points on rank one normal homogeneous spaces. Ann. Glob. Anal. Geom. 45, 211–231 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grama, L., Martins, R.M.: Global behavior of the Ricci flow on generalized flag manifolds with two isotropy summands. Indag. Math. 23, 95–104 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962)zbMATHGoogle Scholar
  11. 11.
    Itoh, M.: On curvature properties of Kähler C-spaces. J. Math. Soc. Jpn. 30(1), 39–71 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 2. Interscience, Wiley, Nova Iorque (1969)zbMATHGoogle Scholar
  13. 13.
    Kowalski, O., Szenthe, J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedic. 81, 209–214 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Bolletino U.M.I 7(5–B), 189–246 (1991)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lafuente, R., Lauret, J.: Structure of homogeneous Ricci solitons and the Alekseevskii conjecture. J. Differ. Geom. 98, 315–347 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lauret, J.: Curvature flows for almost hermitian Lie groups. Trans. Am. Math. Soc. 367, 7453–7480 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Negreiros, C.J.C.: Some remarks about harmonic maps into flag manifolds. Indiana Univ. Math. J. 37(3), 617–636 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Negreiros, C.J.C., Grama, L., San Martin, L.A.B.: Invariant Hermitian structures and variational aspects of a family of holomorphic curves on flag manifolds. Ann. Glob. Anal. Geom. 40, 105–123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    San Martin, L.A.B., Negreiros, C.J.C.: Invariant almost Hermitian structures on flag manifolds. Adv. Math. 178, 277–310 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    San Martin, L.A.B.: Álgebras de Lie. Editora Unicamp, Campinas (2003)Google Scholar
  21. 21.
    Ziller, W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259, 351–358 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ziller, W.: The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces. Comment. Math. Helv. 52, 573–590 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics - IMECCUniversity of CampinasCampinasBrazil
  2. 2.Federal Institute of Brasília - IFB - Campus GamaBrasíliaBrazil

Personalised recommendations