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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149121 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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| _version_ | 1862625574913048576 |
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| author | Alekseevsky, D.V. Nikonorov, Y.G. |
| author_facet | Alekseevsky, D.V. Nikonorov, Y.G. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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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| first_indexed | 2025-12-07T13:35:21Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149121 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:35:21Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
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| spelling | 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 |
| spellingShingle | Compact Riemannian Manifolds with Homogeneous Geodesics Alekseevsky, D.V. Nikonorov, Y.G. |
| title | 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_short | Compact Riemannian Manifolds with Homogeneous Geodesics |
| title_sort | compact riemannian manifolds with homogeneous geodesics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149121 |
| work_keys_str_mv | AT alekseevskydv compactriemannianmanifoldswithhomogeneousgeodesics AT nikonorovyg compactriemannianmanifoldswithhomogeneousgeodesics |