Positively curved $4$-manifolds and the nonnegativity of isotropic curvatures.

Noronha, Maria Helena

doi:10.1307/mmj/1029005700

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Introduction3

Decomposition and Evolution Of Curvature Tensors1

It Has Been Conjectured That1

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Cited by 12 publications

(12 citation statements)

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“…As it was pointed out in [9] the sum of two sectional curvatures on two orthogonal planes appeared in works of Noronha [18] and Seaman [22]. We remark that the positivity of the biorthogonal curvature is an intermediate condition between positive sectional curvature and positive scalar curvature.…”

Section: Introductionsupporting

confidence: 59%

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

Rufino¹

2017

Glasgow Math. J.

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Section: Introductionsupporting

confidence: 59%

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

Rufino¹

2017

Glasgow Math. J.

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“…It is known that in dimension 4, positive isotropic curvature is equivalent to positive Weitzenböck operator (see for example, [17,18,20,21]). For an even dimensional Riemannian manifold of n > 4, positive isotropic curvature implies positive Weitzenböck operator ([24, Proposition 1.1]).…”

Section: Decomposition and Evolution Of Curvature Tensorsmentioning

confidence: 99%

Curvature pinching estimate and singularities of the Ricci flow

Cao

2011

Communications in Analysis and Geometry

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Abstract. In this paper, we first derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and Weyl tensor under the Ricci flow. Then we apply this estimate to study finite-time singularity behavior. We show that if the scalar curvature is uniformly bounded, then the Weyl tensor has to blow up, as a consequence, the corresponding singularity model must be Ricci flat with non-vanishing Weyl tensor.

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“…Corollary 6.26 [9] and [26]). For more details on this subject we address to [3], [7], [21], [22] and [24].…”

Section: It Has Been Conjectured Thatmentioning

confidence: 99%

“…[11], [12] and [16]), nonnegative or positive isotropic curvature (cf. [4], [5], [20], [22] and [25]) and nonnegative or positive biorthogonal curvature (cf. [3], [7] and [24]).…”

Section: Introductionmentioning

confidence: 99%

Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

Ribeiro

2016

Annali di Matematica

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ABSTRACT. The goal of this paper is to investigate the rigidity of 4-dimensional manifolds involving some pinching curvature conditions. To this end, we make use of the approach of biorthogonal curvature which is weaker than the sectional curvature. Here, we prove a rigidity result for 4-dimensional compact manifolds under a suitable lower bound condition on the minimum of the biorthogonal curvature. From this, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional manifolds.

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Positively curved $4$-manifolds and the nonnegativity of isotropic curvatures.

Cited by 12 publications

References 0 publications

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

Curvature pinching estimate and singularities of the Ricci flow

Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

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