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The aim of this paper is to describe some geometrical aspects of Riemannian manifolds with a pole. A point o of a Riemannian manifold is called a pole, if the exponential map exp is a diffeomorphism at o. Simply connected complete Riemannian manifolds of nonpositive curvature (the Euclidean space, the hyperbolic space and a simply connected symmetric space of noncompact type, etc.) and a paraboloid of revolution are typicalexamples of Riemannian manifolds with a pole. We give in I a sufficientcondition on the existence of a pole in terms of curvature. Hessian comparison theorem, conformal changes of a metric and a generalization of Cartan's fixed point theorem are discussed in 2 ([6], [1] C2]). And we argue in 3 the order of a holomorphic function on a Kahler manifold with a pole ([7]).

Some geometrical aspects of Riemannian monifolds with a pole