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Hyperpolar actions and \(k\)-flat homogeneous spaces. (English) Zbl 0804.53074
A closed, connected, \(k\)-dimensional submanifold of a compact Riemannian manifold \(M\) is called a \(k\)-flat of \(M\) if it is flat in the induced metric and totally geodesic. We call \(M\) “\(k\)-flat homogeneous” if every geodesic lies in some \(k\)-flat of \(M\), and if the group of isometries of \(M\) acts transitively on pairs \((\sigma,p)\) consisting of a \(k\)-flat \(\sigma\) and a point \(p\in \sigma\). An isometric action on \(M\) is called hyperpolar if there exists a connected, closed, flat submanifold of \(M\) that meets all orbits orthogonally. We prove that the following three properties for a compact Riemannian manifold \(M\) are equivalent: (a) \(M\) is a Riemannian homogeneous manifold and admits a cohomogeneity \(k\) hyperpolar action with a fixed point, (b) \(M\) is \(k\)- flat homogeneous, (c) \(M\) is a rank \(k\) symmetric space.
Since 1-flat homogeneous is trivially equivalent to two-point homogeneous, the equivalence of (b) and (c) generalizes the well-known fact that two-point homogeneous spaces are the same as rank 1 symmetric spaces.
Reviewer: E.Heintze

53C35 Differential geometry of symmetric spaces
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