A Note on pinching sphere theorem

Yuliang Wen

doi:10.1016/j.crma.2003.12.007

Differential Geometry

A Note on pinching sphere theorem

Presented by: Thierry Aubin

Yuliang Wen ¹

¹ Department of Mathematics, East China Normal University, 20062 Shanghai, PR China

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

Abstract
Résumé

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 $ℝ^{2 n + 1}$ . 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 $(\frac{3}{2} + η) V (S^{2 n})$ 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 $ℝ^{2 n + 1}$ . Nous prouvons que si la courbure sectionnelle K_M varie dans ]0,1] et si le volume V(M) est inférieur à $(\frac{3}{2} + η) V (S^{2 n})$ pour un nombre positif η dependant seulement de n, alors M²ⁿ est homéomorphe à S²ⁿ.

Received: 2003-11-05
Accepted: 2003-12-02
Published online: 2004-01-09

DOI: 10.1016/j.crma.2003.12.007

Author's affiliations:

Yuliang Wen ¹

¹ Department of Mathematics, East China Normal University, 20062 Shanghai, PR China

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     author = {Yuliang Wen},
     title = {A {Note} on pinching sphere theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {229--234},
     publisher = {Elsevier},
     volume = {338},
     number = {3},
     year = {2004},
     doi = {10.1016/j.crma.2003.12.007},
     language = {en},
}

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Yuliang Wen. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.007/

References
Cited by

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

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

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

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

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

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

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

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