On the injectivicy radius growth of complete noncompact Riemannian manifolds. (English) Zbl 1370.53034

Sun, Zhongyang; Wan, Jianming

Asian J. Math. 18, No. 3, 419-426 (2014).

Summary: In this paper we introduce a global geometric invariant \(\alpha(M)\) related to the injectivity radius of complete non-compact Riemannian manifolds and prove: If \(\alpha(M^n) > 1\), then \(M^n\) is isometric to \(\mathbb{R}^n\) when the Ricci curvature is non-negative, and is diffeomorphic to \(\mathbb{R}^n\) for \(n \neq 4\) and homeomorphic to \(\mathbb{R}^4\) for \(n = 4\) without any curvature assumption.

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

injectivity radius; complete non-compact manifold