For a given Riemannian manifold $$(M^n, g)$$ which is near standard sphere $$(S^n, g_{round})$$ in the Gromov–Hausdorff topology and satisfies $$Rc \ge n-1$$ , it known by Cheeger–Colding theory that M diffeomorphic to $$S^n$$ . A diffeomorphism $$\varphi : \rightarrow S^n$$ was constructed Cheeger Colding (J Differ Geom 46(3):406–480, 1997) using Reifenberg method. In this note, we show desire...