Manifolds without conjugate points and with integral curvature zero
The author studies complete noncompact Riemannian manifolds without conjugate points and obtains some generalizations of E. Hopf and L. W. Green's result that a compact manifold without conjugate points has nonpositive integral of the scalar curvature, which vanishes only if a manifold is flat. Namely, the author proves the following theorem: ``Let M be a complete Riemannian manifold without conjugate points. Suppose that the set \(\Omega\) of nonwandering points of the geodesic flow \(f^ t\) on SM (the sphere tangent bundle over M) decomposes into at most countably many \(f^ t\)-invariant sets each of which has finite volume and the Ricci curvature is summable on SM. Then, the integral of scalar curvature of M is nonpositive, and it vanishes only if M is flat.'' The previous statement is true without the assumption on the set \(\Omega\) if M is simply connected (Corollary 1), M has finite volume (Corollary 2), or M is flat outside some compact set (Corollary 3).
scientific article; zbMATH DE number 4115456