A geometric characterization of negatively curved locally symmetric spaces
The author proves a rank 1 version of an earlier higher rank rigidity result obtained independently by W. Ballmann and K. Burns - R. Spatzier. Theorem. Let M be a compact Riemannian manifold with sectional curvature \(K\leq -1\). Assume that for each unit vector \(v\in T_ 1\tilde M\) there is a parallel, perpendicular vector field E(t) on the corresponding geodesic \(\gamma_ V\) such that \(K(E,\gamma_ v')(t)=-1\) for all \(t\in {\mathbb{R}}\). Then M is locally symmetric of rank 1. The rigidity result of Ballmann and Burns-Spatzier can be formulated in a similar way by replacing -1 by zero in the statement above. In the opinion of the reviewer these two results should be regarded as two halves of one more general result although the proofs of the two results are quite different. The common link is the rigidity caused by the presence of parallel, perpendicular vector fields along each geodesic that determine 2-planes with maximum sectional curvatures. Essentially the same dichotomy arises in the Mostow Rigidity Theorem, where the proof falls into two cases: a rank 1 case involving quasiconformal analysis and a higher rank case involving Tits buildings. We outline some ideas of the proof of the author's theorem. Let M denote the universal cover of M and let \(\tilde M(\infty)\) denote the boundary sphere of \(\tilde M.\) 1) For each vector \(v\in T_ 1\tilde M\) the author introduces a non- Riemannian metric \(\eta_ v\) on \(M(\infty)-\{\gamma_ v(-\infty)\}\), a set which can be identified naturally with the strong unstable horosphere \(W^{su}(v)\subseteq T_ 1M\) determined by v. The metrics \(\eta_ v\) depend on v but are 1-quasiconformally equivalent on \(\tilde M(\infty).\) 2) For each \(v\in T_ 1M\) and each \(\xi \in T_ v(T_ 1\tilde M)\), the vector field \(Y_{\xi}(t)=(d\pi \circ g^ t)(\xi)\) is a Jacobi vector field on \(\gamma_ v\), where \(\{g^ t\}\) denotes the geodesic flow in \(T_ 1\tilde M\) and \(\pi: T_ 1\tilde M\to \tilde M\) is the projection. Let \(E_ v=\{\xi \in T_ v(T_ 1\tilde M):\) \(E_{\xi}(t)/| Y_{\xi}(t)|\) is a parallel, perpendicular vector field on \(\gamma_ v\) with \(K(E_{\xi},\gamma_ v')(t)=-1\) for all \(t\in {\mathbb{R}}\}\). The hypothesis of the theorem implies that dim \(E_ v\geq 1\) for all \(v\in T_ 1\tilde M\). The ergodicity of the geodesic flow on the compact quotient manifold M implies that there exists a subset A of \(T_ 1M\) with full measure in \(T_ 1\tilde M\) such that for all \(v\in A\) the restriction of \(E_ v\) to \(W^{su}(v)\) is a Carnot-Caratheodory distribution on \(W^{su}(v).\) 3) The author shows that for all \(v\in A\) the corresponding Carnot- Caratheodory metric \(\delta_ v\), defined on \(W^{su}(v)\) by the distribution \(E_ v\) and the Riemannian metric of \(W^{su}(v)\), equals a metric \(\ell_ v\) defined purely in terms of the metric \(\eta_ v\) from 1). Using results of M. Gromov and J. Mitchell the author proves that there is a simple connected nilpotent Lie group N with a left invariant Carnot-Caratheodory metric \(\delta\) such that for all \(v\in T_ 1\tilde M\) the metric space \((W^{su}(v),\lambda_ v)\) is isometric to \((N,\delta)\). 4) The distances \(\{\ell_ v:\) \(v\in T_ 1M\}\) determine a well defined 1-quasiconformal structure on \(\tilde M(\infty)\). If G denotes the group of 1-quasiconformal homeomorphisms of \(\tilde M(\infty)\), then G acts transitively on \(\tilde M(\infty)\) by 3). In the case that \(\tilde M\) is symmetric, G is the restriction to \(\tilde M(\infty)\) of \(I(\tilde M)\), the isometry group of \(\tilde M.\) In the present situation the author's task is to show that G can be extended to act as a transitive group of isometries of M. Since \(\tilde M\) admits a compact quotient M it then follows from a result of E. Heinze that \(\tilde M\) is symmetric of rank 1. 5) The first step in this task is to extend G to a transitive group of homeomorphisms of \(T_ 1\tilde M\) that commute with the geodesic flow. The next step is to show that each element of G acting on \(T_ 1\tilde M\) preserves the fibers of \(\pi: T_ 1\tilde M\to \tilde M\). It then follows that G induces a transitive group of homeomorphisms of \(\tilde M\) that are easily seen to be isometries.
scientific article; zbMATH DE number 4212923