Skip to main content
SpringerLink
Log in

Geodesic orbit spaces of compact Lie groups of rank two

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H, g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agricola, I., Ferreira, A.C., Friedrich, T.: The classification of naturally reductive homogeneous spaces in dimensions \(n\le 6\). Differ. Geom. Appl. 39, 59–92 (2015)

    Article  Google Scholar 

  2. Alekseevsky, D.V., Arvanitoyeorgos, A.: Riemannian flag manifolds with homogeneous geodesics. Trans. Am. Math. Soc. 359, 3769–3789 (2007)

    Article  MathSciNet  Google Scholar 

  3. Alekseevsky, D.V., Nikonorov, Y.G.: Compact Riemannian manifolds with homogeneous geodesics. SIGMA Symmetry Integrability Geom. Methods Appl. 5(093), 16 pages (2009)

  4. Arvanitoyeorgos, A.: Homogeneous manifolds whose geodesics are orbits: recent results and some open problems. Irish Math. Soc. Bull. 79, 5–29 (2017)

    Article  MathSciNet  Google Scholar 

  5. Berndt, J., Kowalski, O., Vanhecke, L.: Geodesics in weakly symmetric spaces. Ann. Global Anal. Geom. 15, 153–156 (1997)

    Article  MathSciNet  Google Scholar 

  6. Berstovskii, V.N., Nikonorov, Y.G.: On \(\delta \)-homogeneous Riemannian manifolds. Differ. Geom. Appl. 26, 514–535 (2008)

    Article  MathSciNet  Google Scholar 

  7. Berstovskii, V.N., Nikonorov, Y.G.: Clifford–Wolf homogeneous Riemannian manifolds. J. Differ. Geom. 82, 467–500 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Berstovskii, V.N., Nikonorov, Y.G.: Generalized normal homogeneous Riemannian metrics on spheres and projective spaces. Ann. Global Anal. Geom. 45, 167–196 (2014)

    Article  MathSciNet  Google Scholar 

  9. Borel, A., De Siebenthal, J.: Les sous-groupes fermés de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23, 200–221 (1949)

    Article  MathSciNet  Google Scholar 

  10. Calvaruso, G., Zaeim, A.: Four-dimensional pseudo-Riemannian g.o. spaces and manifolds. J. Geom. Phys. 130, 63–80 (2018)

  11. Chen, H., Chen, Z., Wolf, J.: Geodesic orbit metrics on compact simple Lie groups arising from flag manifolds. C. R. Math. 356, 846–851 (2018)

    Article  MathSciNet  Google Scholar 

  12. Chen, Z., Nikonorov, Y.G.: Geodesic orbit Riemannian spaces with two isotropy summands I. Geom. Dedicata. 203, 163–178 (2019)

    Article  MathSciNet  Google Scholar 

  13. D’Atri, J. E., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Am. Math. Soc. 18(215) (1979)

  14. Douglas, A., Repka, J.: Levi decomposable subalgebras of the symplectic algebra \(C_2\). J. Math. Phys. 56, 051703 (2015)

    Article  MathSciNet  Google Scholar 

  15. Douglas, A., Repka, J.: Subalgebras of the rank two semisimple Lie algebras. Linear Multilinear Algebra 66, 2049–2075 (2018)

    Article  MathSciNet  Google Scholar 

  16. Dynkin, E. B.: Semisimple subalgebras of semisimple Lie algebras. Mat. Sb. (N.S.) 30 (72) no. 2, 349–462 (1952)

  17. Gordon, C.S.: Homogeneous Riemannian manifolds whose geodesics are orbits, In: Topics in Geometry. Progress in Nonlinear Differential Equations and Their Applications, vol 20, Birkhäuser, Boston (1996)

  18. Gordon, C., Nikonorov, Y.G.: Geodesic orbit Riemannian structures on \(R^n\). J. Geom. Phys. 134, 235–243 (2018)

    Article  MathSciNet  Google Scholar 

  19. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978)

    MATH  Google Scholar 

  20. Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Unione Mat. Ital. B 5(7), 189–246 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Krämer, M.: Eine klassifikation bestimmter untergruppen kompakter zusammenhängender Liegruppen. Commun. Algebra 3, 691–737 (1975)

    Article  Google Scholar 

  22. Manturov, O.V.: Homogeneous asymmetric Riemannian spaces with an irreducible group of motions. Dokl. Akad. Nauk SSSR 141, 792–795 (1961)

    MathSciNet  Google Scholar 

  23. Manturov, O.V.: Riemannian spaces with orthogonal and symplectic groups of motions and an irreducible group of rotations. Dokl. Akad. Nauk SSSR 141, 1034–1037 (1961)

    MathSciNet  MATH  Google Scholar 

  24. Manturov, O.V.: Homogeneous Riemannian manifolds with irreducible isotropy group. Trudy Sere. Vector. Tenzor. Anal. 13, 68–145 (1966)

    Google Scholar 

  25. Mayanskiy, E.: The subalgebras of \(G_2\). arXiv:1611.04070v1 (2016). https://arxiv.org/pdf/1611.04070v1.pdf

  26. Nikonorov, Yu.G.: Geodesic orbit Riemannian metrics on spheres. Vladikavkaz. Mat. Zh. 15(3), 67–76 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Nikonorov, Yu.G.: On the structure of geodesic orbit Riemannian spaces. Ann. Global Anal. Geom. 52, 289–311 (2017)

    Article  MathSciNet  Google Scholar 

  28. Nomizu, K.: Studies on Riemannian homogeneous spaces. Nagoya Math. J. 9, 43–56 (1955)

    Article  MathSciNet  Google Scholar 

  29. Olmos, C., Reggiani, S., Tamaru, H.: The index of symmetry of compact naturally reductive spaces. Math. Z. 277, 611–628 (2014)

    Article  MathSciNet  Google Scholar 

  30. Souris, N.P.: Geodesic orbit metrics in compact homogeneous manifolds with equivalent isotropy submodules. Transform. Groups 23, 1149–1165 (2018)

    Article  MathSciNet  Google Scholar 

  31. Souris, N.P.: On a class of geodesic orbit spaces with abelian isotropy subgroup. Manuscripta Math. 166, 101–129 (2021)

    Article  MathSciNet  Google Scholar 

  32. Storm, R.: The classification of 7- and 8-dimensional naturally reductive spaces. Can. J. Math. 72, 1246–1274 (2020)

    Article  MathSciNet  Google Scholar 

  33. Tamaru, H.: Riemannian g.o. spaces fibered over irreducible symmetric spaces. Osaka J. Math. 36, 835–851 (1999)

  34. Wang, M., Ziller, W.: On isotropy irreducible Riemannian manifolds. Acta Math. 166, 223–261 (1991)

    Article  MathSciNet  Google Scholar 

  35. Wolf, J.A.: The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math. 120, 59–148 (1968)

    Article  MathSciNet  Google Scholar 

  36. Wolf, J.A.: The geometry and structure of isotropy irreducible homogeneous spaces. correction Acta Math. 152, 141–142 (1984)

  37. Wolf, J.A.: Harmonic Analysis on Commutative Spaces. Mathematical Surveys and Monographs, Vol. 142, American Mathematical Society, Providence, RI (2007)

  38. Yan, Z., Deng, S.: Finsler spaces whose geodesics are orbits. Differ. Geom. Appl. 36, 1–23 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This research is co-financed by Greece and the European Union (European Social Fund - ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning 2014-2020” in the context of the project “Geodesic orbit metrics on homogeneous spaces of classical Lie groups” (MIS 5047124).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Patras, University Campus, 26504, Rio Patras, Greece

    Nikolaos Panagiotis Souris

Authors
  1. Nikolaos Panagiotis Souris
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolaos Panagiotis Souris.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Souris, N.P. Geodesic orbit spaces of compact Lie groups of rank two. Geom Dedicata 216, 1 (2022). https://doi.org/10.1007/s10711-021-00668-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10711-021-00668-1

Keywords

Mathematics Subject Classification 2010

Search

Navigation