Compact Riemannian Manifolds with Homogeneous Geodesics
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 invaria...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2009 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2009
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/149121 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Compact Riemannian Manifolds with Homogeneous Geodesics / D.V. Alekseevsky, Y.G. Nikonorov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-149121 |
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dspace |
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Alekseevsky, D.V. Nikonorov, Y.G. 2019-02-19T17:31:47Z 2019-02-19T17:31:47Z 2009 Compact Riemannian Manifolds with Homogeneous Geodesics / D.V. Alekseevsky, Y.G. Nikonorov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 28 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53C20; 53C25; 53C35 https://nasplib.isofts.kiev.ua/handle/123456789/149121 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 M₁ = SO(2n+1)/U(n) or M₂ = Sp(n)/U(1)·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 g₀ such that (M,g0) is the symmetric space M = SO(2n+2)/U(n+1) or, respectively, CP²n⁻¹. The manifolds M₁, M₂ are weakly symmetric spaces. This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. The first author was partially supported by the Royal Society (Travel Grant 2007/R3). The second author was partially supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (grant NSH-5682.2008.1). We are grateful to all referees, whose comments and suggestions permit us to improve the presentation of this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Compact Riemannian Manifolds with Homogeneous Geodesics Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Compact Riemannian Manifolds with Homogeneous Geodesics |
| spellingShingle |
Compact Riemannian Manifolds with Homogeneous Geodesics Alekseevsky, D.V. Nikonorov, Y.G. |
| title_short |
Compact Riemannian Manifolds with Homogeneous Geodesics |
| title_full |
Compact Riemannian Manifolds with Homogeneous Geodesics |
| title_fullStr |
Compact Riemannian Manifolds with Homogeneous Geodesics |
| title_full_unstemmed |
Compact Riemannian Manifolds with Homogeneous Geodesics |
| title_sort |
compact riemannian manifolds with homogeneous geodesics |
| author |
Alekseevsky, D.V. Nikonorov, Y.G. |
| author_facet |
Alekseevsky, D.V. Nikonorov, Y.G. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 M₁ = SO(2n+1)/U(n) or M₂ = Sp(n)/U(1)·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 g₀ such that (M,g0) is the symmetric space M = SO(2n+2)/U(n+1) or, respectively, CP²n⁻¹. The manifolds M₁, M₂ are weakly symmetric spaces.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149121 |
| citation_txt |
Compact Riemannian Manifolds with Homogeneous Geodesics / D.V. Alekseevsky, Y.G. Nikonorov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 28 назв. — англ. |
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AT alekseevskydv compactriemannianmanifoldswithhomogeneousgeodesics AT nikonorovyg compactriemannianmanifoldswithhomogeneousgeodesics |
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2025-12-07T13:35:21Z |
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2025-12-07T13:35:21Z |
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1850856724383137793 |