Compact Riemannian Manifolds with Homogeneous Geodesics

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2009
• Hauptverfasser: Alekseevsky, D.V., Nikonorov, Y.G.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2009
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/149121
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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.

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.