manuscripta math. 31, 297 - 316 (1980) manuscripta mathematica 9 by Spring~-Verl~ 1980 LINE INTEGRATION OF RICCI CURVATURE AND CONJUGATE POINTS IN LORENTZIAN AND RIEMANNIAN MANIFOLDS Carmen Chicone and Paul Ehrlich* Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian mani- folds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspaceUke geo- desic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incom- pleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds. 0. Introduction Generalizing a result of Cohn-Vossen [9] for the Gauss curvature, Gromoll and Meyer [16] showed using index form techniques that if c : ~ -* (M, g) is a complete geodesic in a Riemannian manifold with Ric(c'(t), c'(t)) >0 for all t ~ ~ and Ric(c'(to) , c'(to)) > 0 for some t c ~, then c contains a pair of conjugate points. Using this result, o Gromoll and Meyer [16, p. 80] proved that a complete noncompact Riemannian manifold with everywhere positive Ricci curvature has only one end. % Partially supported by NSF grant MCS77-18723(02). 0025-2611/80/0031/0297/$04.00 297