Classification of manifolds with weakly 1/4-pinched curvatures

Brendle, Simon; Schoen, Richard

doi:10.1007/s11511-008-0022-7

Cited by 119 publications

(158 citation statements)

References 15 publications

(38 reference statements)

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“…In 1991, Chen [9] showed that a point-wise 1/4-pinched 4-manifold is diffeomorphic to a spherical space form. Recently, Brendle and Schoen [10,11] proved a remarkable differentiable pinching theorem for point-wise 1/4-pinched Riemannian manifolds by developing the theory and techniques of Ricci flow [22][23][24]. More recently, Brendle [12] obtained the following useful result.…”

Section: Some Useful Lemmasmentioning

confidence: 99%

Topological and differentiable sphere theorems for complete submanifolds

Zhao

2009

Communications in Analysis and Geometry

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We investigate topological and differentiable structures of submanifolds under extrinsic restrictions. We first obtain a topological sphere theorem for compact submanifolds in a Riemannian manifold. Secondly, we prove an optimal differentiable sphere theorem for 4-dimensional complete submanifolds in a space form, which provides a partial solution of the smooth Poincaré conjecture. Finally, we prove some new differentiable sphere theorems for n-dimensional submanifolds in a Riemannian manifold.

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Section: Some Useful Lemmasmentioning

confidence: 99%

Topological and differentiable sphere theorems for complete submanifolds

Zhao

2009

Communications in Analysis and Geometry

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“…Moreover, Brendle and Schoen [7] obtained a differentiable rigidity theorem for compact manifolds with weakly 1/4-pinched curvatures in the pointwise sense. Using Brendle and Schoen's result [7], Petersen and Tao [28] proved a classification theorem for compact and simply connected manifolds with almost 1/4-pinched sectional curvatures.…”

Section: Introductionmentioning

confidence: 99%

Geometric, topological and differentiable rigidity of submanifolds in space forms

2013

Geom. Funct. Anal.

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Let M be an n-dimensional submanifold in the simply connected space form F n+p (c) with c + H 2 > 0, where H is the mean curvature of M . We verify that if M n (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric M ≥ (n − 2)(c + H 2 ), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or CP 2 (sphere. We then prove that if M n (n ≥ 4) is a compact submanifold in F n+p (c) with c ≥ 0, and if Ric M > (n − 2)(c + H 2 ), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.

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“…These are not necessarily quarter-pinched (as this property is not preserved by Ricci flow), but they do have nonnegative isotropic curvatures when we add a factor of R 2 to the metric. Thus we can use the classification in [6] to understand what the candidates for limit manifolds are. Finally it is worth mentioning that there is at least one example of an exotic sphere with positive sectional curvature (see [14]).…”

Section: Theorem 11 There Exist ε (N) > 0 So That Any Simply Connecmentioning

confidence: 99%

“…Below we will show that the product metric on M × R 2 has nonnegative isotropic curvature for t > 0. The Brendle-Schoen classification (see [6,Theorem 2]) of such metrics then shows that M and hence M i (for sufficiently large i) are diffeomorphic to a sphere or compact rank one symmetric space, giving the required contradiction.…”

Section: Theorem 11 There Exist ε (N) > 0 So That Any Simply Connecmentioning

confidence: 99%