In 1961, S. Kobayashi [4] proved the following: Theorem. Let M be a complete Riemannian manifold. If there exists a point p of M such that no geodesic passing through p contains a point conjugate to p, then the universal covering space of M is diffeomorphic to Euclidean space. More precisely, the exponential map exp,: Mp-+M is a covering map. This theorem had been proved by Myers [6] under the assumption that M was analytic, and this assumption was essential in his proof. Kobayashi, and also Helgason in his book [3], showed that analyticity was superfluous and could be replaced with mere smoothness of a sufficiently high order. We want to consider this theorem for infinite-dimensional Riemannian Hubert manifolds. We do not know if it is true in unrestricted generality; the introduction in §3 of an ad hoc assumption on the metric is necessary for our method to work. But the class of manifolds satisfying the assumption is large, including all negatively curved manifolds as well as some which are everywhere positively curved. The main results are stated in Theorems 4.1 and 4.4. Our definition of completeness is as follows: a manifold M is complete if it is Cauchy complete in its intrinsic topological metric. Unfortunately, infinite-dimensional manifolds complete in this sense may be incomplete in another sense: they may bear two points unconnectable by a minimal geodesic (see Example 5.1). The question of conjugate points arises. They may be defined in the finite-dimensional case as singularities of the exponential map. We define them similarly in the infinite-dimensional case but find in §2 that two types appear, which we call monoconjugate and epiconjugate. The epiconjugate are the more sensitive measure of pathology. §5 contains examples demonstrating that pathological distributions of conjugate points can occur. One effect of the ad hoc assumption of §3 is to rule out the possibility of conjugate points. The author wishes to thank W. Klingenberg who brought Kobayashi's paper [4] to his attention while the writing of this present
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