Classification of almost quarter-pinched manifolds

Petersen, Peter; Tao, Terence

doi:10.1090/s0002-9939-09-09802-5

Cited by 31 publications

(19 citation statements)

References 15 publications

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“…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. The following important convergence result for the Ricci flow in higher dimensions, initiated by Brendle and Schoen [6] and finally verified by Brendle [4], cut open a new field in curvature and topology of manifolds [5,8,37].…”

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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Section: Introductionmentioning

confidence: 99%

Geometric, topological and differentiable rigidity of submanifolds in space forms

2013

Geom. Funct. Anal.

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“…Then M is either diffeomorphic to S n or isometric to a compact rank-one symmetric space (CROSS). Petersen and Tao [2009] improved Brendle and Schoen's pinching constant in Theorem A to 1 if M is an n-dimensional compact Riemannian manifold (n ≥ 3) whose sectional curvature and scalar curvature satisfy K M ≤ 1 and R > n(n − 1)δ n , then M is diffeomorphic to a spherical space form. Here δ n = 1 − 20 − 8 sgn(n − 3) 5n(n − 1) and sgn( · ) is the standard sign function.…”

Section: Introductionmentioning

confidence: 99%

A differentiable sphere theorem inspired by rigidity of minimal submanifolds

Xu¹,

Tian²

2011

Pacific J. Math.

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We prove a vanishing theorem for the fundamental group of a compact submanifold in a space form, and then present a refined version of Ejiri's rigidity theorem for minimal submanifolds in a sphere. Inspired by the refined Ejiri theorem, we verify a new differentiable sphere theorem for compact submanifolds in space forms. We also show that our differentiable sphere theorem is sharp. We emphasize that our method of Ricci flow in the proof of the sphere theorem seems useful in the study of curvature and topology. Also, we obtain a differentiable pinching theorem for compact submanifolds in a Riemannian manifold.

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“…[BöWi08,BrSc09,BrSc08,NiWo07,PeTa09]). An important step in the original proofs involves estimating the injectivity radii of manifolds with quarter pinched positive sectional curvatures.…”

Section: Introductionmentioning

confidence: 99%