Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

Ribeiro, E.

doi:10.1007/s10231-016-0557-8

Cited by 9 publications

(8 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the sequel, as an application of Theorem 2 combined with results by Freedman [15] and Donaldson [14], we get the following corollary. It may be interesting to compare Corollary 1 with Theorem 1.3 in [26]. In fact, our latter result requires the same pinching condition of Theorem 1.3 in [26]; however, it does not require conditions on the Weyl tensor and analyticity of the metric.…”

Section: Theorem 1 Let (M 4 G) Be a Four-dimensional Compact Orienmentioning

confidence: 87%

“…In particular, they showed that a four-dimensional compact oriented Einstein manifold 1 10 -pinched is either topologically S 4 or homothetically isometric to CP 2 . Besides, the main result in [26] implies that a four-dimensional compact Einstein manifold M 4 with normalized Ricci curvature Ric = 1 and sectional curvature K ≥ 1 12 must be isometric to either S 4 or CP 2 ; see also [9,12,33]. Indeed, it remains a challenging task to obtain new classification results under weaker curvature pinching conditions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Four-Manifolds With Positive Curvature

Diógenes¹,

Ribeiro²,

Rufino³

2020

Glasgow Math. J.

View full text Add to dashboard Cite

We prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M 4 is topologically S 4 or CP 2 , provided that the sectional curvatures all lie in the interval [ 3 √ 3−5 4, 1]. In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

show abstract

Section: Theorem 1 Let (M 4 G) Be a Four-dimensional Compact Orienmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Four-Manifolds With Positive Curvature

Diógenes¹,

Ribeiro²,

Rufino³

2020

Glasgow Math. J.

View full text Add to dashboard Cite

show abstract

“…where K ⊥ max (p) = max{K ⊥ (P); P ⊂ T p M}. More details can be found in [9,21] and references therein.…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, it is not difficult to check that (λμ) 2 − 2λμ + 1 ≥ 0, for all λ, μ, which settles our claim (26). Proceeding, from inequalities (25), (21) and (26) we obtain…”

Section: Proof Of Theoremmentioning

confidence: 99%

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

Rufino¹

2017

Glasgow Math. J.

View full text Add to dashboard Cite

show abstract

“…Furthermore, he showed that

S^{4}

♯^{m} {double-struckCP}^{2} ♯^{n} {\bar{CP}}^{2}

and

♯^{n} ({double-struckS}^{2} \times {double-struckS}^{2})

admit metrics with positive biorthogonal curvature. For more details on this subject, see, for instance and .…”

Section: Introductionmentioning

confidence: 99%

Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4‐manifolds

Costa

Ribeiro

2018

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds.In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded. K E Y W O R D S 4-manifold, biorthogonal curvature, minimal volume, sphere theorem M S C ( 2 0 1 0 ) Primary: 53C21, 53C23; Secondary: 53C25This concept plays an important role in geometric topology. It is closely related with other important invariants as, for instance, minimal entropy h(M) and simplicial volume ‖ ‖. Paternain and Petean [37] proved that the minimal volume, on a compact manifold , satisfies the following chain of inequalitieswhere ( ) is a positive constant; for more details, we refer the reader to [23,24] and [37]. It should be emphasized that some authors have studied other minimal volume invariants in a similar context (cf. [30,45] and [46]). Among them, let us highlight the following ones: Gromov minimal volume, which is defined by

show abstract

Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

Cited by 9 publications

References 26 publications

Four-Manifolds With Positive Curvature

Four-Manifolds With Positive Curvature

Some Sphere Theorems for Submanifolds With Positive Biorthogonal Curvature

Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4‐manifolds

Contact Info

Product

Resources

About