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Communicated by Stephen Smale, June 11, 1968 0. In this paper we describe some results on the structure of complete manifolds of nonnegative sectional curvature. (We will denote such manifolds by M.) Details and related results will appear elsewhere. Our work generalizes that of Gromoll, Meyer [2] as well as Toponogov [4].Our analysis is divided into two parts. The first, which we feel is of particular interest, essentially reduces the study of the topology of complete manifolds of nonnegative curvature to that of compact manifolds of nonnegative curvature. The second step reduces the compact case to the compact simply connected case (modulo the classification of compact flat manifolds).Our results have various applications. One of these is the classification up to isometry of complete 3-dimensional manifolds of nonnegative curvature.

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The interest in periodic geodesies arose at a very early stage of differential geometry, and has grown rapidly since then. It is a basic general problem to estimate the number of distinct periodic geodesies c: R -• Λί, c(t + 1) = c(t) 9 on a complete riemannian manifold M in terms of topological invariants. Here periodic geodesies are always understood to be non-constant, and two such curves c l9 c 2 will be said to be distinct if they are geometrically different, c x (R) Φ c£R).If M is non-compact, then it is even difficult to find reasonable conditions for the existence of at least one periodic geodesic, see §4. However, in the compact case many results are available. It is classical and rather elementary that any non-trivial conjugacy class of the fundamental group n x {M) gives rise to periodic geodesies, and very often the existence of a larger number of distinct periodic geodesies can be deduced from further properties of π x (M) compare [4, p. 240], [7], and also §4. When M is simply connected* the problem is getting much more delicate. Here the first result was obtained in 1905 by Poincare, who proved that there exists a periodic geodesic on every surface analytically equivalent to the euclidean sphere S2 . Yet rather late, in 1952, Fet and Lusternik established the theorem that on any compact riemannian manifold M at least one geodesic is periodic. Several authors have proved the existence of certain finite numbers of distinct periodic geodesies for special topological types of manifolds, partly under restrictive metric conditions. We mention the work of Lusternik, Schnirelmann, Morse, Fet, Alber, and Klingenberg; for references see [11].In §4 of this paper we shall prove: There always exist infinitely many distinct periodic geodesies on an arbitrary compact manifold, provided some weak topological condition holds. In fact, it seems that our condition will be satisfied except in comparatively few cases. Until now it was not even possible to decide whether there is some compact simply connected differentiable manifold M with infinitely many distinct periodic geodesies for all riemannian structures on M.As yet the mast natural ami successful way of dealing with periodic geodesies

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