Geodesics without conjugate points and curvatures at infinity

Abstract

We study the asymptotic behavior of curvature and prove that the integral of curvature along a geodesic without conjugate points is nonpositive and some generalizations of Myers theorem and Cohn-Vossen's theorem. Some applications are also given.Key words: Riemannian manifold, geodesic, conjugate point. MAIN RESULTSLet M n be an n-dimensional Riemannian manifold and let d(x, y) be the distance induced by the metric. (Ambrose 1957) showed that if the integral of the Ricci curvature along a geodesic γ : [0, +∞) → M is infinite then there is a t > 0 such that γ (t) is conjugate to γ (0). We extended this result in two directions: first we obtain geodesic without conjugate points (particularly if γ is a line). Then for any unit vector field X(t) which is parallel along γ it holds that where K X(t), γ (t) is the sectional curvature of the plane spanned by X(t) and γ (t). MoreoverNote that M is not supposed to be complete in Theorem A, and no hypothesis on the curvature is assumed. It should be remarked that (Liang & Zhan 1996) proved that if γ : R → M is a geodesic without conjugate points and if Ric γ (t), γ (t) ≥ 0, then Ric γ (t), γ (t) ≡ 0.