Positive Ricci curvature on fiber bundles with compact structure group

F., Cavenaghi, Leonardo; Sperança, Llohann D.

doi:10.1515/advgeom-2021-0007

Cited by 7 publications

(8 citation statements)

References 18 publications

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Section: Polar Foliations Toric Symmetry and Calabi-yau Bundlesmentioning

confidence: 99%

The Symmetric Kazdan–warner Problem and Applications

Sperança

2022

Preprint

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After R. Schoen completed the solution of the Yamabe problem, compact manifolds could be categorized in three classes, depending on whether they admit a metric with positive, non-negative, or only negative scalar curvature. Here we follow Yamabe’s first attempt to solve his problem through variational methods and provide an analogous equivalent classification for manifolds equipped with actions by non-discrete compact Lie groups. Moreover, we apply the method and the results to: classify total spaces of fiber bundles with compact structure groups (concerning scalar curvature), to conclude density results and compare realizable scalar curvature functions between some exotic manifolds and their standard counterpart. We also provide an extended range of prescribed scalar curvature functions of warped products, especially with Calabi–Yau manifolds.

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Section: Polar Foliations Toric Symmetry and Calabi-yau Bundlesmentioning

confidence: 99%

The Symmetric Kazdan–warner Problem and Applications

Sperança

2022

Preprint

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“…Example 1 (The Gromoll-Meyer exotic sphere). This construction first appeared in [GM72] and was first put in a ⋆-diagram in [Dur01] (see also [CS19]). Consider the compact Lie group ) ′ yields (S 7 ) ′ diffeomorphic to the connected sum of k times Σ 7 GM ; (Σ 8 ): there is a S 3 -equivariant suspension η : S 8 → S 7 of the Hopf map S 3 → S 2 whose quotient (S 8 ) ′ = η * Sp(2)/S 3 is the only exotic 8-sphere; (Σ 10 ): there is a S 3 -equivariant suspension θ : S 10 → S 7 of a generator of π 6 S 3 whose induced ⋆-quotient (S 10 ) ′ is a generator of the index 2 subgroup os homotopy 10-spheres that bound spin manifolds;…”

Section: Equivariant Constructions and Examplesmentioning

confidence: 99%

The symmetric Kazdan-Warner problem and applications

F.¹,

do²,

Sperança³

2021

Preprint

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After R. Schoen completed the solution of the Yamabe problem, compact manifolds could be allocated in three classes depending on whether they admit a metric with positive, non-negative or only negative scalar curvature. Here we follow Yamabe's first attempt to solve his problem through variational methods and provide an analogous equivalent classification for manifolds equipped with actions by non-discrete compact Lie groups. Moreover, we apply the method, and the results to classify total spaces of fibre bundles with compact structure groups (concerning scalar curvature), to conclude density results, and compare realizable scalar curvature functions between some exotic manifolds their standard counterpart. We also provide an extended range of prescribed scalar curvature functions of warped products, especially with Calabi-Yau manifolds, providing an upper bound for the first positive eigenvalue of the Laplacian under relatively mild conditions.

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“…We now prove Theorem D and generalize it to several bundles. These constructions are deeply relied on the ones in [CS19].…”

Section: More Particularlymentioning

confidence: 99%

“…Nash ([Nas79]), Poor ( [Poo75]), ), Wraith, Joachim and Crowley ([Wra97,Wra07], [JW08] and [CW17a,CW17b]) proved the existence of metrics of positive Ricci curvature on some exotic manifolds. In [CS18,CS19], the authors built metrics of positive Ricci curvature on several exotic manifolds and bundles with fibers and/or bases of exotic manifolds. On the other hand, it is now known is there exists an exotic sphere with a metric of positive sectional curvature and Hitchin proved that there are exotic spheres that do not even admit metrics of positive scalar curvature (see [Hit74]).…”

Section: Introductionmentioning

confidence: 99%