Differential Geometry
A Note on pinching sphere theorem
Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 229-234.

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 ${ℝ}^{2\mathrm{n}+1}$. We prove in this note that if the sectional curvature KM varies in (0,1] and the volume V(M) is not larger than $\left(\frac{3}{2}+\eta \right)\mathrm{V}\left({\mathrm{S}}^{2\mathrm{n}}\right)$ for some positive number η depending only on n, then M2n is homeomorphic to S2n.

Soit M2n une variété riemannienne compacte, simplement connexe de dimension 2n sans bord et soit S2n la sphère unitée de l'espace euclidien ${ℝ}^{2\mathrm{n}+1}$. Nous prouvons que si la courbure sectionnelle KM varie dans ]0,1] et si le volume V(M) est inférieur à $\left(\frac{3}{2}+\eta \right)\mathrm{V}\left({\mathrm{S}}^{2\mathrm{n}}\right)$ pour un nombre positif η dependant seulement de n, alors M2n est homéomorphe à S2n.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.12.007
Wen, Yuliang 1

1 Department of Mathematics, East China Normal University, 20062 Shanghai, PR China
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Wen, Yuliang. A Note on pinching sphere theorem. Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 229-234. doi : 10.1016/j.crma.2003.12.007. http://www.numdam.org/articles/10.1016/j.crma.2003.12.007/

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