On complete manifolds of nonnegative 𝑘th-Ricci curvature

Shen, Zhong Min

doi:10.1090/s0002-9947-1993-1112548-6

Cited by 35 publications

(6 citation statements)

References 28 publications

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“…The geometry and topology of k-th Ricci curvature was initiated by Hartman [20] in 1979, and developed by Wu [42] and Shen [36,37], etc.. By the definition above, it is seen that the Ricci curvature of M is equal to the (n − 1)-th Ricci curvature, (n − 1, 1)-curvature and 1-th weak Ricci curvature; the scalar curvature of M is equal to (n − 1, n)-curvature, n-th weak Ricci curvature and (n − 1)-th scalar curvature. For any unit tangent vector…”

Section: For Any Unit Tangent Vectormentioning

confidence: 99%

The sphere theorems for manifolds with positive scalar curvature

Gu¹,

Xu²

2012

J. Differential Geom.

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Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if M n is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R 0 > σ n K max , where σ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if M n (n ≥ 3) is a compact manifold whose (n − 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric (n−2) min > τ n (n−2)R 0 , where τ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker and the authors. * 2010 Mathematics Subject Classification. 53C20; 53C40. exists for all time and converges to a constant curvature metric as t → ∞. Here r g(t) denotes the mean value of the scalar curvature of g(t).Theorem B([10]). Let (M, g 0 ) be a compact, locally irreducible Riemannian manifold of dimension n(≥ 4). Assume that M × R 2 has nonnegative isotropic curvature, i.e.,for all orthonormal four-frames {e 1 , e 2 , e 3 , e 4 } and all λ, µ ∈ [−1, 1]. Then one of the following statements holds: (i) M is diffeomorphic to a spherical space form.(ii) n = 2m and the universal cover of M is a Kähler manifold biholomorphic to CP m . (iii) The universal cover of M is isometric to a compact symmetric space.On the other hand, some important work on sphere theorems for manifolds with positive Ricci curvature have been made by several geometers (see [3, 14, 21, 30, 36, 39], etc.). In 1990's, Cheeger, Colding and Petersen [14, 30] proved the following differentiable sphere theorem for manifolds with positive Ricci curvature.Theorem C. Let M n be a compact and simply connected Riemannian n-manifold with Ricci curvature Ric M ≥ n − 1. Suppose that one of the following conditions holds:where ω n = vol(S n ) and ε 1 (n) is some positive constant; (ii) λ n+1 < n + ε 2 (n), where λ n+1 is the (n + 1)-th eigenvalue of M and ε 2 (n) is some positive constant. Then M is diffeomorphic to S n . Let K(π) be the sectional curvature of M for 2-plane π ⊂ T x M , Ric(u) the Ricci curvature of M for unit vector u ∈ U x M . Set K max (x) := max π⊂TxM K(π), Ric min (x) := min u∈UxM Ric(u). Inspired by Shen's topological sphere theorem [36], the authors [44] obtained the following differentiable sphere theorem for manifolds of positive Ricci curvatures.Theorem D. Let M n be a compact Riemannian n-manifold. If Ric min > δ n (n − 1)K max , where δ n = 1 − 6 5(n−1) , then M is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to S n . Let M n be a submanifold in a Riemannian manifold M N . Denote by H and S the mean curvature and the s...

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Section: For Any Unit Tangent Vectormentioning

confidence: 99%

The sphere theorems for manifolds with positive scalar curvature

Gu¹,

Xu²

2012

J. Differential Geom.

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“…where we use the assumption (1) in (20). This together with Theorem A proves that M is diffeomorphic to a space form.…”

Section: Sphere Theorems For Submanifolds Of Positive K-th Ricci Curvmentioning

confidence: 68%

“…Theorem B ( [31,34]) Let M n be an n-dimensional oriented complete submanifold in the simply connected space form F n+ p (c) with nonnegative constant curvature c. The geometry and topology of k-th Ricci curvature was initialed by P. Hartman [11] in 1979, and developed by Wu [29], Shen [20] and others. We recall the definition of the k-th Ricci curvature.…”

Section: Introductionmentioning

confidence: 99%

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

2011

manuscripta math.

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We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M n is simply connected and the k-th Ricci curvature of M n is bounded below by a quantity involving the mean curvature of M n and the curvature of the ambient manifold, then M n is diffeomorphic to the standard sphere S n . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M n is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.

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“…Let γ be a minimal geodesic from p to q. If Ric (k) (x) ≥ 0 on all x of M , Abresch-Gromoll [2] (for k = n − 1) and Shen [8](for any k) proved that…”

Section: Prelimanariesmentioning

confidence: 99%

Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth

Zhang

2015

Proc. Amer. Math. Soc.

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In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generalize it to the manifolds with kth asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate.

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On complete manifolds of nonnegative 𝑘th-Ricci curvature

Abstract: Abstract.In this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature. Key tools are comparison techniques and Morse Theory of Busemann and distance functions.

Cited by 35 publications

References 28 publications

The sphere theorems for manifolds with positive scalar curvature

The sphere theorems for manifolds with positive scalar curvature

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth

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