On Manifolds All of Whose Geodesics are Closed

Abstract

By a closed geodesic g of length w on a Riemann manifold M is meant a map g: R -> M of the reals [oc 0. g is called simple if the arc g(x) (O 0, then M is simply connected, and its integral cohomology ring is a truncated polynomial ring, generated by an element 0 of dimension X + 1. A truncated polynomial ring is a ring obtained from Z(x), the ring of polynomials over the integers, by adding the single relation xn = 0. The restrictions imposed on H(M) by virtue of this theorem are quite strong. For instance: X + 1 must divide dim M; if X + 1 is odd, then X + 1 = dim M; the Poincare polynomial of M over any field K is of the form P(t) = 1 + t+1 + t2(X+) ... tk(X+). Much more subtle restrictions on a truncated polynomial ring which is the cohomology ring of a complex have recently been obtained by J. Adem [5]. For instance, it follows from his work that dim 0 is a power of two, whenever 02 5z 0, and if dim 0 > 8, then O' = 0. The only known simply connected manifolds whose cohomology ring is of the type under discussion are the following ones: A. The spheres S'; (n _ 2); B. The complex projective spaces Zn; (n _ 1); C. The quaternion projective spaces Qf; (n > 1); D. The Cayley projective plane C2 . These spaces are precisely the irreducible symmetric spaces of rank 1 of Cartan [4]. Conversely, Cartan's classification shows that amongst symmetric spaces the irreducible ones of rank 1 are the only ones which have their geodesics simply closed at some point. Cartan's work therefore gives a much stronger result than our theorem under the strong additional condition that M be symmetric. 375