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SIGMA, 2009, Volume 5, 093 (Mi sigma439)  

This article is cited in 17 scientific papers (total in 17 papers)

Compact Riemannian Manifolds with Homogeneous Geodesics

Dmitrii V. Alekseevskiia, Yurii G. Nikonorovb

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

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 $M_1=SO(2n+1)/U(n)$ or $M_2=Sp(n)/U(1)\cdot Sp(n-1)$ 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 $g_0$ such that $(M,g_0)$ is the symmetric space $M=SO(2n+2)/U(n+1)$ or, respectively, $\mathbb{C}P^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces.

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

DOI: https://doi.org/10.3842/SIGMA.2009.093

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Bibliographic databases:

ArXiv: 0904.3592
Document Type: Article
MSC: 53C20; 53C25; 53C35
Received: April 22, 2009; in final form September 20, 2009; Published online September 30, 2009
Language: English

Citation: Dmitrii V. Alekseevskii, Yurii G. Nikonorov, “Compact Riemannian Manifolds with Homogeneous Geodesics”, SIGMA, 5 (2009), 093, 16 pp.

Citation in format AMSBIB
\Bibitem{AleNik09}
\by Dmitrii V.~Alekseevskii, Yurii G.~Nikonorov
\paper Compact Riemannian Manifolds with Homogeneous Geodesics
\jour SIGMA
\yr 2009
\vol 5
\papernumber 093
\totalpages 16
\mathnet{http://mi.mathnet.ru/sigma439}
\crossref{https://doi.org/10.3842/SIGMA.2009.093}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84896060374}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Tóth G.Z., “On Lagrangian and Hamiltonian systems with homogeneous trajectories”, J. Phys. A, 43:38 (2010), 385206, 19 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    2. Cohen N., Grama L., Negreiros C.J.C., “Equigeodesics on flag manifolds”, on J. Math., 37:1 (2011), 113–125  mathscinet  zmath  isi
    3. Berestovskii V.N., Nikitenko E.V., Nikonorov Yu.G., “Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic”, Differential Geom Appl, 29:4 (2011), 533–546  crossref  mathscinet  isi  elib  scopus
    4. Jovanovic B., “Geodesic Flows on Riemannian g.o. Spaces”, Regular & Chaotic Dynamics, 16:5 (2011), 504–513  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Yu. G. Nikonorov, “Geodesic orbit Riemannian metrics on spheres”, Vladikavk. matem. zhurn., 15:3 (2013), 67–76  mathnet  zmath
    6. Nikonorov Yu.G., “Geodesic Orbit Manifolds and Killing Fields of Constant Length”, Hiroshima Math. J., 43:1 (2013), 129–137  mathscinet  zmath  isi  elib
    7. Berestovskii V.N., Gorbatsevich V.V., “Homogeneous Spaces With Inner Metric and With Integrable Invariant Distributions”, Anal. Math. Phys., 4:4 (2014), 263–331  crossref  mathscinet  zmath  isi  elib  scopus
    8. Yan Z., Deng Sh., “Finsler Spaces Whose Geodesics Are Orbits”, Differ. Geom. Appl., 36 (2014), 1–23  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Nikonorov Yu.G., “Killing Vector Fields of Constant Length on Compact Homogeneous Riemannian Manifolds”, Ann. Glob. Anal. Geom., 48:4 (2015), 305–330  crossref  mathscinet  zmath  isi  elib  scopus
    10. Nikolayevsky Y., “Totally Geodesic Hypersurfaces of Homogeneous Spaces”, Isr. J. Math., 207:1 (2015), 361–375  crossref  mathscinet  zmath  isi  scopus
    11. Berestovskii, V. N.; Gorbatsevich, V. V., “Homogeneous spaces with inner metric and with integrable invariant distributions”, Journal of Mathematical Sciences (United States), 207:3 (2015), 410-466  crossref  zmath  scopus
    12. Arvanitoyeorgos A., Wang Yu., Zhao G., “Riemannian G.O. Metrics in Certain M-Spaces”, Differ. Geom. Appl., 54:A (2017), 59–70  crossref  mathscinet  zmath  isi  scopus
    13. Nikonorov Yu.G., “On the Structure of Geodesic Orbit Riemannian Spaces”, Ann. Glob. Anal. Geom., 52:3 (2017), 289–311  crossref  mathscinet  zmath  isi  scopus
    14. Arvanitoyeorgos A., Wang Yu., “Homogeneous Geodesics in Generalized Wallach Spaces”, Bull. Belg. Math. Soc.-Simon Steven, 24:2 (2017), 257–270  mathscinet  zmath  isi
    15. Dusek Z., “Homogeneous Geodesics and G.O. Manifolds”, Note Mat., 38:1 (2018), 1–15  crossref  isi  scopus
    16. Calvaruso G., Zaeim A., “Four-Dimensional Pseudo-Riemannian G.O. Spaces and Manifolds”, J. Geom. Phys., 130 (2018), 63–80  crossref  mathscinet  zmath  isi  scopus
    17. Chen H., Chen Zh., Wolf J.A., “Geodesic Orbit Metrics on Compact Simple Lie Groups Arising From Flag Manifolds”, C. R. Math., 356:8 (2018), 846–851  crossref  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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