Inventiones mathematicae

, Volume 140, Issue 2, pp 423-452

First online:

Complete Riemannian manifolds with pointwise pinched curvature

  • Bing-Long ChenAffiliated withDepartment of Mathematics, Zhongshan University, Guangzhou, P.R. China
  • , Xi-Ping ZhuAffiliated withDepartment of Mathematics, Zhongshan University, Guangzhou, P.R. China

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Abstract.

An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold M n . We prove that if n≥4 and the curvature operator of M n is pointwise pinched, or if n=3 and the Ricci curvature of M 3 is pointwise pinched, then M n is compact.