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A review of compact geodesic orbit manifolds and the g.o. condition for \(\operatorname{SU}(5)/\operatorname{S(U}(2) \times \operatorname{U}(2))\). (English) Zbl 1501.53049

In a Riemannian g.o. space (g.o. manifold), every geodesic is the orbit of a 1-parameter group of isometries. A classification of g.o. spaces is known only in low dimensions or for special isometry groups, or special isotropy groups, respectively. For a compact isometry group \(G\), one of the techniques for the study of g.o. spaces is based on the existence of an invariant scalar product on the Lie algebra \({\mathfrak{g}}\). Then any g.o. metric is related with this invariant scalar product by a symmetric endomorphism and the study of g.o. metrics reduces to the study of these endomorphisms.
In the present paper, a review of selected results on g.o. spaces with compact isometry group \(G\) is presented. Further, the special g.o. space \({\mathrm{SU}}(5)/{\mathrm{S}}({\mathrm{U}}(2)\times{\mathrm{U}}(2))\) is studied.

MSC:

53C22 Geodesics in global differential geometry
53C30 Differential geometry of homogeneous manifolds
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