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# Asian Journal of Mathematics

## Volume 18 (2014)

### Number 3

### On the injectivity radius growth of complete non-compact Riemannian manifolds

Pages: 419 – 426

DOI: http://dx.doi.org/10.4310/AJM.2014.v18.n3.a2

#### Authors

#### Abstract

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

#### Keywords

injectivity radius, complete non-compact manifold

#### 2010 Mathematics Subject Classification

Primary 53C20. Secondary 53C35.