Manifolds with 1/4-pinched curvature are space forms

Brendle, Simon; Schoen, Richard

doi:10.1090/s0894-0347-08-00613-9

Cited by 277 publications

(323 citation statements)

References 29 publications

(63 reference statements)

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“…Motivated by the result above and the striking differentiable pinching theorem due to Brendle and Schoen [2009], we propose the following question.…”

Section: Questionsmentioning

confidence: 99%

L^pRicci curvature pinching theorems for conformally flat Riemannian manifolds

Xu¹,

Zhao²

2010

Pacific J. Math.

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Dedicated to Professor Katsuhiro Shiohama on the occasion of his 70th birthday.Let M be an n-dimensional complete locally conformally flat Riemannian manifold with constant scalar curvature R and n ≥ 3. We first prove that if R = 0 and the L n/2 norm of the Ricci curvature tensor of M is pinched in [0, C 1 (n)), then M is isometric to a complete flat Riemannian manifold, which improves Pigola, Rigoli, and Setti's pinching theorem. Next, we prove that if n ≥ 6, R = 0, and the L n/2 norm of the trace-free Ricci curvature tensor of M is pinched in [0, C 2 (n)), then M is isometric to a space form. Finally, we prove an L n trace-free Ricci curvature pinching theorem for complete locally conformally flat Riemannian manifolds with constant nonzero scalar curvature. Here C 1 (n) and C 2 (n) are explicit positive constants depending only on n.

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“…Motivated by the result above and the striking differentiable pinching theorem due to Brendle and Schoen [2009], we propose the following question.…”

Section: Questionsmentioning

confidence: 99%

L^pRicci curvature pinching theorems for conformally flat Riemannian manifolds

Xu¹,

Zhao²

2010

Pacific J. Math.

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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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“…Recently Böhm and Wilking [3] proved that every manifold with 2-positive curvature operator must be diffeomorphic to a space form. More recently, Brendle and Schoen [6] proved the remarkable differentiable sphere theorem for manifolds with pointwise 1/4-pinched 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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Manifolds with 1/4-pinched curvature are space forms

Cited by 277 publications

References 29 publications

L^pRicci curvature pinching theorems for conformally flat Riemannian manifolds

L^pRicci curvature pinching theorems for conformally flat Riemannian manifolds

Topological and differentiable sphere theorems for complete submanifolds

Geometric, topological and differentiable rigidity of submanifolds in space forms

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Manifolds with 1/4-pinched curvature are space forms

Cited by 277 publications

References 29 publications

LpRicci curvature pinching theorems for conformally flat Riemannian manifolds

LpRicci curvature pinching theorems for conformally flat Riemannian manifolds

Topological and differentiable sphere theorems for complete submanifolds

Geometric, topological and differentiable rigidity of submanifolds in space forms

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Product

Resources

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L^pRicci curvature pinching theorems for conformally flat Riemannian manifolds

L^pRicci curvature pinching theorems for conformally flat Riemannian manifolds