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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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