## ABSTRACT

Many basic properties of complete, noncompact Riemannian manifolds stem from the principle that a limit curve of a sequence of minimal geodesics is itself a minimal geodesic. After the correct formulation of completeness had been given by Hopf and Rinow (1931), Rinow (1932) and Myers (1935) were able to establish the existence of a geodesic ray issuing from every point of a complete noncompact Riemannian manifold using this principle. Here a geodesic γ : [0, ∞) → (N, g
_{0}) is said to be a ray if γ realizes the Riemannian distance between every pair of its points. Rinow and Myers constructed the desired geodesic ray as follows. Since (N, g
_{0}) is complete and noncompact, there exists an infinite sequence {p_{n}
} of points in N such that for every point p ∈ N, d
_{0}(p , P_{n}
) → ∞ as n → ∞. Let γ_{n}
be a minimal (i.e., distance realizing) unit speed geodesic segment from p = γ_{n}
(0) to p_{n}
. This segment exists by the completeness of (N, g
_{0}). If v ∈ T_{p}N is any accumulation point of the sequence {γ_{n}′(0)} of unit tangent vectors in T_{p}N, then γ(t) = exp
_{p} tv is the required geodesic ray. Intuitively, γ is a ray since it is a limit curve of some subsequence of the minimal geodesic segments {γ_{n}
}. The existence of geodesic rays through every point has been an essential tool in the structure theory of both positively curved [cf. Cheeger and Gromoll (1971, 1972)] and negatively curved [cf. Eberlein and O’Neill (1973)] complete noncompact Riemannian manifolds.