A Note on pinching sphere theorem

Let M²ⁿ be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S²ⁿ be the unit sphere of 2n+1 dimension Euclidean space $\text{R}{}^{\mathrm{2n+1}}$. We prove in this note that if the sectional curvature K_M varies in (0,1] and the volume V(M) is not larger than $\text{(}\text{3}\text{2}\text{+η)V(S}{}^{\mathrm{2n}}\text{)}$ for some positive number η depending only on n, then M²ⁿ is homeomorphic to S²ⁿ. To cite this article: Y. Wen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).