Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 093, 16 pages      arXiv:0904.3592
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Compact Riemannian Manifolds with Homogeneous Geodesics

Dmitrii V. Alekseevsky a and Yurii G. Nikonorov b
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

Received April 22, 2009, in final form September 20, 2009; Published online September 30, 2009

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.

Key words: homogeneous spaces; weakly symmetric spaces; homogeneous spaces of positive Euler characteristic; geodesic orbit spaces; normal homogeneous Riemannian manifolds, geodesics.

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