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Entropy-rigidity of locally symmetric spaces of negative curvature. (English) Zbl 0699.53049

Let \(S\) be a compact locally symmetric space of negative curvature, i.e. a compact quotient of a hyperbolic space \(\mathbb{K}H^n\) where \(\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}\) or \(\mathbb{K}=Ca\) and \(n=2\). Suppose that the symmetric metric on \(\mathbb{K}H^n\) is normalized such that the maximum of its sectional curvature \(K\) is \(-1\). From the rigidity theorem of G. D. Mostow [Strong rigidity of locally symmetric spaces. Princeton, N. J.: Princeton University Press (1973; Zbl 0265.53039)] it results that if two normalized locally symmetric spaces are homotopy equivalent then they are isometric. It follows that the topological entropy \(h(S)\) of the geodesic flow on the unit tangent bundle \(T^1S\) of \(S\) represents an invariant of the homotopy type of \(S\).
In this paper the following interesting result is proved: Let \(M\) be a compact manifold \((\dim M\ge 3)\) which is homotopy equivalent to the compact locally symmetric space \(S\). If the curvature of \(M\) does not exceed \(-1\) then \(h(M)\ge h(S)\). Moreover, \(h(M)=h(S)\) implies that \(M\) and \(S\) are isometric. The proof of the above result is technical, and it uses some ideas from the following two papers of P. Pansu [“Géométrie conforme grossière” (Preprint), and Ann. Math. (2) 129, No. 1, 1–60 (1989; Zbl 0678.53042)].

MSC:

53C20 Global Riemannian geometry, including pinching
53C35 Differential geometry of symmetric spaces
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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