Periodic Metrics

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 P on a complete manifold M possessing an isometry group r with a compact quotient M/f. The word "global" means here that we study '·large" objects and do not care of the measurement error of order diam(M/f). We consider here only a special (but rather natural) case when P is a perturbation (not necessarily small) of a constant curvature metric Po with the same group of isometries. We have two main possibilities: the flat case when Po is Euclidean metric on M ~ IR n and f ~ zn acts by integer translations, and the hyperbolic case when r is a hyperbolic group. We denote geometric structures attached to Po by marking with a circle like the followings: expo-the exponential map, < .,. >0_ the inner product, UOTM-the unit tangent bundle etc. This paper is the continuation of [2] , and here we give its abstract. Definition 1. For (x, v) E UOTM we define K(x,v) = limt_x tp(x,exp~tv) if the limit exists. It is simple to check that K depends on the infinity point (which is a class of asymptotic geodesics) [-y(t) = exp~ tv] E a O AI only, and we denote also K[/,] = K(x, v) for [/,] = [-y(t) = exp~ tv] E a O M.