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.
KeywordsRiemannian Metrics Hyperbolic Group Isometric Embedding Geodesic Flow Finsler Manifold
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