1994/01/01 by Dmitri Burago · 2 citations
Mathematics · #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Mathematics and Applications
paper · doi:10.1007/978-3-0348-7515-8_7
openalex publication_date 1994/01/01 · openalex created_date 2022/05/12 · openalex updated_date 2026/08/04
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 ≈ ℝ n and Γ ≈ ℤ n 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: exp 0 — the exponential map, < ∙, ∙ > 0 — the inner product, U 0 TM- the unit tangent bundle etc. 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.