A Note on pinching sphere theorem

Wen, Yuliang

Comptes Rendus. Mathématique, Tome 338 (2004) no. 3, pp. 229-234.

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 K_M varie dans ]0,1] et si le volume V(M) est inférieur à $(\frac{3}{2}+\eta )\mathrm{V}\left({\mathrm{S}}^{2\mathrm{n}}\right)$ pour un nombre positif η dependant seulement de n, alors M²ⁿ est homéomorphe à S²ⁿ.

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

[1] Cheeger, J.; Ebin, D.G. Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975

[2] Coghlan, L.; Itokawa, Y. A sphere theorem for reverse volume pinching on even-dimension manifolds, Proc. Amer. Math. Soc., Volume 111 (1991) no. 3, pp. 815-819

[3] Gallot, S.; Hulin, D.; Lafontaine, J. Riemannian Geometry, Springer-Verlag, 1993

[4] Grove, K.; Shiohama, K. A generalized sphere theorem, Ann. of Math., Volume 106 (1977), pp. 201-211

[5] Sakai, T. Riemannian Geometry, Shokabo, Tokyo, 1992

[6] Shiohama, K. A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc., Volume 27 (1983), pp. 811-819

[7] Wang, B.F. A topological sphere theorem for volume on even-dimensional compact Riemannian manifold, J. Sichuan Univ. Natur. Sci. Ed., Volume 34 (1997) no. 5, pp. 581-583 (in Chinese)

[8] Wu, J.Y. A diameter pinching sphere theorem, Proc. Amer. Math. Soc., Volume 197 (1990) no. 3, pp. 796-802