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

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} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2559668} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000271092200029} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84896060374} 

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This publication is cited in the following articles:
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12. Arvanitoyeorgos A., Wang Yu., Zhao G., “Riemannian G.O. Metrics in Certain M-Spaces”, Differ. Geom. Appl., 54:A (2017), 59–70
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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
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