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

This article is cited in 11 scientific papers (total in 11 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
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
\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. 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
    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
    4. Jovanovic B., “Geodesic Flows on Riemannian g.o. Spaces”, Regular & Chaotic Dynamics, 16:5 (2011), 504–513  crossref  mathscinet  zmath  adsnasa  isi
    5. Yu. G. Nikonorov, “Geodesic orbit Riemannian metrics on spheres”, Vladikavk. matem. zhurn., 15:3 (2013), 67–76  mathnet
    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
    8. Yan Z., Deng Sh., “Finsler Spaces Whose Geodesics Are Orbits”, Differ. Geom. Appl., 36 (2014), 1–23  crossref  mathscinet  zmath  adsnasa  isi  elib
    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
    10. Nikolayevsky Y., “Totally Geodesic Hypersurfaces of Homogeneous Spaces”, Isr. J. Math., 207:1 (2015), 361–375  crossref  mathscinet  zmath  isi
    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
  • Symmetry, Integrability and Geometry: Methods and Applications
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