Abstract
Let M be a homogeneous Riemannian manifold and G the isometry group of M. Then M can be viewed as a coset space G/H with a left-invariant Riemannian metric. M is said to be a g. o. manifold if every geodesic in M is an orbit of a one-parameter subgroup of G. In [KV1], O. Kowalski and L. Vanhecke showed that every g. o. manifold is a D’Atri space; i. e. the local geodesic symmetries are volume preserving up to sign. (See the survey article by Kowalski, Prufer and Vanhecke in this volume for a discussion of D’Atri spaces and additional comments about g. o. manifolds.) The simplest Riemannian homogeneous spaces are the naturally reductive manifolds; these form a subclass of the g. o. manifolds. The definition of naturally reductive (see §1) involves a purely algebraic condition on the isometry group.
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© 1996 Birkhäuser Boston
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Gordon, C.S. (1996). Homogeneous Riemannian Manifolds Whose Geodesies Are Orbits. In: Gindikin, S. (eds) Topics in Geometry. Progress in Nonlinear Differential Equations and Their Applications, vol 20. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2432-7_4
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