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Abstract

The topic which served as the starting point for this investigation is the “global” geometry of periodic metrics. We call periodic a Riemannian metric ρ on a complete manifold M possessing an isometry group Γ with a compact quotient M/Γ. The word “global” means here that we study “large” objects and do not care of the measurement error of order diam(M/Γ). We consider here only a special (but rather natural) case when ρ is a perturbation (not necessarily small) of a constant curvature metric ρ 0 with the same group of isometries. We have two main possibilities: the flat case when ρ 0 is Euclidean metric on M ≈ ℝn and Γ ≈ ℤn acts by integer translations, and the hyperbolic case when Γ is a hyperbolic group. We denote geometric structures attached to ρ 0 by marking with a circle like the followings: exp0 — the exponential map, < ∙, ∙ >0 — the inner product, U 0 TM- the unit tangent bundle etc.

Keywords

Riemannian Metrics Hyperbolic Group Isometric Embedding Geodesic Flow Finsler Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • D. Burago

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