## A differential geometric characterization of symmetric spaces of higher rank.(English)Zbl 0722.53041

A simply connected complete Riemannian manifold M of nonpositive curvature (Hadamard manifold) is of higher rank if every geodesic lies in a totally geodesic flat subspace (“flat”) of dimension $$\geq 2$$. The only known irreducible examples of such manifolds are symmetric spaces of higher rank. Ballmann and Burns/Spatzier have shown that these are in fact the only possible examples provided that (i) the curvature is bounded from below and (ii) there exists a discrete subgroup of the isometry group I(M) such that its orbit space has finite volume. Using a different proof, the present paper generalizes this result in two ways: The authors omit the assumption (i) and replace (ii) by the more general duality condition for I(M) saying that for any geodesic $$\gamma$$, there exists a sequence $$(\phi_ n)$$ in I(M) such that $$\phi_ n(p)\to \gamma (\infty)$$ and $$\phi_ n^{-1}(p)\to \gamma (-\infty)$$ for any $$p\in M.$$
One of the main tools of the proof is taken from a previous paper of the first author, based on an idea of Ballmann: If the duality condition holds, the orbits of the holonomy group, projected to M($$\infty)$$, are contained in the closures of isometry orbits on M($$\infty)$$. Thus, by Berger’s holonomy theorem, it is enough to find a closed nonempty proper subset $$A\subset M(\infty)$$ which is invariant under I(M). It is shown that $$A=M_{\alpha}(\infty)$$ for some $$\alpha\in (0,\pi)$$ does the job, where $M_{\alpha}(\infty)=\{x\in M(\infty);\quad y\in M(\infty);\quad p\in M;\quad\sphericalangle_ p(x,y)=\alpha \}$ and $$\sphericalangle_ p(x,y)$$ is the angle between the geodesics from p to x and to y. To show that $$M_{\alpha}(\infty)\neq \phi$$, the authors use the Tits distance at M($$\infty)$$ which was introduced by Gromov. The main observation is (Lemma 4.2): If $$v\in T^ 1M$$ is a regular recurrent vector tangent to a flat F, then the Tits distance ball $$B_{\alpha}(x)$$ for $$x=\gamma \vee (\infty)$$ and for small $$\alpha >0$$ is contained in F($$\infty)$$.

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C35 Differential geometry of symmetric spaces
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### References:

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