Manifolds without conjugate points and with integral curvature zero
type: Item
instance of: scholarly article
title: Manifolds without conjugate points and with integral curvature zero
zbMATH Open document ID: 0681.53026
author: Nobuhiro Innami
published in: Journal of the Mathematical Society of Japan
publication date: 1989-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 4115456
zbMATH Keywords: geodesic
zbMATH Keywords: conjugate points
zbMATH Keywords: scalar curvature
zbMATH Keywords: nonwandering points
zbMATH Keywords: geodesic flow
zbMATH Keywords: simply connected
zbMATH Keywords: finite volume
zbMATH Keywords: flat outside some compact set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04120251
OpenAlex ID: W1993174724
DOI: 10.2969/JMSJ/04120251
description: scientific article; zbMATH DE number 4115456