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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

##### MSC:
 53C35 Differential geometry of symmetric spaces
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