Soit M2n une variété riemannienne compacte, simplement connexe de dimension 2n sans bord et soit S2n la sphère unitée de l'espace euclidien . Nous prouvons que si la courbure sectionnelle KM varie dans ]0,1] et si le volume V(M) est inférieur à 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 . We prove in this note that if the sectional curvature KM varies in (0,1] and the volume V(M) is not larger than for some positive number η depending only on n, then M2n is homeomorphic to S2n.
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Yuliang Wen 1
@article{CRMATH_2004__338_3_229_0, 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}, }
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/
[1] Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975
[2] A sphere theorem for reverse volume pinching on even-dimension manifolds, Proc. Amer. Math. Soc., Volume 111 (1991) no. 3, pp. 815-819
[3] Riemannian Geometry, Springer-Verlag, 1993
[4] A generalized sphere theorem, Ann. of Math., Volume 106 (1977), pp. 201-211
[5] Riemannian Geometry, Shokabo, Tokyo, 1992
[6] A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc., Volume 27 (1983), pp. 811-819
[7] 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] A diameter pinching sphere theorem, Proc. Amer. Math. Soc., Volume 197 (1990) no. 3, pp. 796-802
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