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Periodic metrics. (English) Zbl 0762.53023
Representation theory and dynamical systems, Adv. Sov. Math. 9, 205-210 (1992).
[For the entire collection see Zbl 0745.00063.]
Let \(\Gamma\) be a cocompact lattice in \(\mathbb{R}^ n\) and let \(\rho\) be a \(\Gamma\)-invariant Riemannian metric on \(\mathbb{R}^ n\). The author shows that for every geodesic line \(\gamma\) in \(\mathbb{R}^ n\) parametrized by arc length the limit \(\lim_{t\to\infty} {1\over t}\rho(\gamma(0),\gamma(t))\) exists and depends only on a point of the ideal boundary of \(\mathbb{R}^ n\) determined by \(\gamma\). Moreover \(\| x\|=| x|\lim_{t\to\infty} {1\over t}\rho(0,tx/| x|)\) defines a norm on \(\mathbb{R}^ n\) with the property that \(\rho\) differs from \(\|\;\|\) by a constant which only depends on the diameter of \((\mathbb{R}^ n/\Gamma,\rho)\) and bounds for the ratio of \(\rho\) and \(\rho_ 0\).

53C20 Global Riemannian geometry, including pinching