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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 $$\rho_ 0$$.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
asymptotic properties; periodic metrics; ideal boundary