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SIGMA, 2009, Volume 5, 093, 16 pp. (Mi sigma439)  

This article is cited in 26 scientific papers (total in 26 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


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

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

Citation in format AMSBIB
\by Dmitrii V.~Alekseevskii, Yurii G.~Nikonorov
\paper Compact Riemannian Manifolds with Homogeneous Geodesics
\jour SIGMA
\yr 2009
\vol 5
\papernumber 093
\totalpages 16

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    This publication is cited in the following articles:
    1. Souris N.P., “On a Class of Geodesic Orbit Spaces With Abelian Isotropy Subgroup”, Manuscr. Math.  crossref  isi
    2. 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
    3. Cohen N., Grama L., Negreiros C.J.C., “Equigeodesics on flag manifolds”, on J. Math., 37:1 (2011), 113–125  mathscinet  zmath  isi
    4. 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
    5. Jovanovic B., “Geodesic Flows on Riemannian g.o. Spaces”, Regular & Chaotic Dynamics, 16:5 (2011), 504–513  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Yu. G. Nikonorov, “Geodesic orbit Riemannian metrics on spheres”, Vladikavk. matem. zhurn., 15:3 (2013), 67–76  mathnet
    7. Nikonorov Yu.G., “Geodesic Orbit Manifolds and Killing Fields of Constant Length”, Hiroshima Math. J., 43:1 (2013), 129–137  mathscinet  zmath  isi  elib
    8. 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
    9. Yan Z., Deng Sh., “Finsler Spaces Whose Geodesics Are Orbits”, Differ. Geom. Appl., 36 (2014), 1–23  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. 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
    11. Nikolayevsky Y., “Totally Geodesic Hypersurfaces of Homogeneous Spaces”, Isr. J. Math., 207:1 (2015), 361–375  crossref  mathscinet  zmath  isi  scopus
    12. 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
    13. 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
    14. 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
    15. Arvanitoyeorgos A., Wang Yu., “Homogeneous Geodesics in Generalized Wallach Spaces”, Bull. Belg. Math. Soc.-Simon Steven, 24:2 (2017), 257–270  mathscinet  zmath  isi
    16. Dusek Z., “Homogeneous Geodesics and G.O. Manifolds”, Note Mat., 38:1 (2018), 1–15  crossref  isi  scopus
    17. 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
    18. 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  mathscinet  zmath  isi  scopus
    19. Xu M., “Geodesic Orbit Spheres and Constant Curvature in Finsler Geometry”, Differ. Geom. Appl., 61 (2018), 197–206  crossref  mathscinet  zmath  isi  scopus
    20. Arvanitoyeorgos A., Wang Yu., Zhao G., “Riemannian M-Spaces With Homogeneous Geodesics”, Ann. Glob. Anal. Geom., 54:3 (2018), 315–328  crossref  mathscinet  zmath  isi  scopus
    21. Gordon C.S., Nikonorov Yu.G., “Geodesic Orbit Riemannian Structures on R-N”, J. Geom. Phys., 134 (2018), 235–243  crossref  mathscinet  zmath  isi  scopus
    22. Nikolayevsky Y., Nikonorov Yu.G., “On Invariant Riemannian Metrics on Ledger-Obata Spaces”, Manuscr. Math., 158:3-4 (2019), 353–370  crossref  mathscinet  zmath  isi  scopus
    23. Nikonorov Yu.G., “On Left-Invariant Einstein Riemannian Metrics That Are Not Geodesic Orbit”, Transform. Groups, 24:2 (2019), 511–530  crossref  isi
    24. Nikonorov Yu.G., “Spectral Properties of Killing Vector Fields of Constant Length”, J. Geom. Phys., 145 (2019), UNSP 103485  crossref  mathscinet  isi  scopus
    25. Chen Zh., Nikonorov Yu., “Geodesic Orbit Riemannian Spaces With Two Isotropy Summands. i”, Geod. Dedic., 203:1 (2019), 163–178  crossref  mathscinet  isi
    26. Arvanitoyeorgos A., Qin H., Wang Yu., Zhao G., “Riemannian Generalized C-Spaces With Homogeneous Geodesics”, Filomat, 33:4, SI (2019), 1117–1124  crossref  mathscinet  isi  scopus
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