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Homogeneous geodesics and g.o. manifolds. (English) Zbl 1401.53041

One interesting problem in pseudo-Riemannian geometry is the description of geodesics. In order to obtain some simplifications to the problem, symmetry conditions on the manifold are assumed. In this framework, the problem of the description of geodesics in homogeneous pseudo-Riemannian manifolds \((M,g)\) is considered.
Motivated by the facts that the geodesics in a Lie group \(G\) with a bi-invariant metric are the one-parameter subgroups of \(G\), and that the geodesics in a Riemannian symmetric space \(G/H\) are orbits of the one-parameter subgroups in \(G/H\), it is natural to look for geodesics in a homogeneous space, which are orbits.
A geodesic \(\gamma (t)\) through the origin \(p\in M\) of a homogeneous pseudo-Riemannian space \((M=G/H,g)\) is said to be homogeneous if it is an orbit of a one-parameter subgroup of \(G\); that is, \[ \gamma (t)=\exp (tX)\cdot p,\quad t\in \mathbb{R}, \] where \(X\) is a non-zero vector in the Lie algebra \(\mathfrak{g}\) of \(G\). The vector \(X\in \mathfrak{g}\) is called a geodesic vector.
A homogeneous pseudo-Riemannian manifold \(M\) all of whose geodesics are homogeneous is called a pseudo- Riemannian g.o.manifold. If the isometry group \(G\) and the presentation \(M=G/H\) is fixed, it is called a g.o. space.
If the metric \(g\) is positive definite, then \((G/H,g)\) is a reductive homogeneous space, but if the metric \(g\) is indefinite, the reductive decomposition may not exist.
Therefore, the techniques in the study of geometrical problems between reductive and non-reductive pseudo-Riemannian manifolds are different. In the reductive case the geodesic vectors are characterized by an algebraic tool called geodesic lemma, meanwhile the non-reductive case can be studied in the broader context of homogeneous affine manifolds using the more fundamental affine method based on Killing vector fields. In Finsler geometry, both the algebraic approach to reductive spaces and the affine approach can be used.
The author presents both the techniques used and the previous and recent results related with the existence of homogeneous geodesics and g.o. manifolds in Riemannian, pseudo-Riemanian, affine and Finsler geometry. The paper under review is a detailed survey on the subject.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C22 Geodesics in global differential geometry
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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