Paper · Nov 9, 2019

Positive Ricci curvature on fiber bundles with compact structure group

Leonardo Francisco Cavenaghi, Llohann Dallagnol Sperança

math.DGarXiv:1911.03608v6
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Abstract

The aim of this paper is to present a direct and simple proof of a result concerning the existence of metrics of positive Ricci curvature on the total space of fiber bundles with compact structure groups. In particular, it also generalizes and puts in a unified framework the results in Nash and Poor. With the intention of disseminating this result, we apply it to build new examples of manifolds with positive Ricci curvature, including bundles whose base consists of gradient shrinking Ricci solitons.

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Submitted
Nov 9, 2019
Last revised
Aug 25, 2020 (v6)
Subjects
math.DG
DOI
10.1515/advgeom-2021-0007

Comments: Exposition substantially improved. Examples added

Ricci solitons

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On Mean curvature flow of Singular Riemannian foliations: Non compact cases (Marcos M. Alexandrino, Leonardo F. Cavenaghi and Icaro Gonçalves, 2019)6Alexandrino 2019 The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds (Lino Grama, Ricardo M. Martins, Mauro Patrão et al., 2020)5Grama 2020 On the embeddability of the homogeneous Ricci flow and its collapses (Mauro Patrão, Lucas Seco and Llohann D. Sperança, 2021)4Patrão 2021 On the dynamics of positively curved metrics on SU(3)/T² under the homogeneous Ricci flow (Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins, 2023)2Cavenaghi 2023 The complete dynamics description of positively curved metrics in the Wallach flag manifold SU(3)/T² (Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al., 2023)1Cavenaghi 2023 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)7Chu 2024 Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature (Steven Gindi, 2024)8Gindi 2024 On a special class of gradient Ricci solitons (José Nazareno Vieira Gomes and Marcus Antonio Mendonça Marrocos, 2026)3Gomes 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The complete dynamics description of positively curved metrics in the Wallach flag manifold

    Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al.

    Same author Leonardo Francisco Cavenaghi · Shared terms “fiber bundles”

  2. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins

    Same author Leonardo Francisco Cavenaghi · Shared terms “positive Ricci”

  3. On a special class of gradient Ricci solitons

    José Nazareno Vieira Gomes and Marcus Antonio Mendonça Marrocos

    Topics Ricci solitons · Shared terms “fiber bundles”, “shrinking Ricci”

  4. On the embeddability of the homogeneous Ricci flow and its collapses

    Mauro Patrão, Lucas Seco and Llohann D. Sperança

    Same author Llohann Dallagnol Sperança

  5. The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds

    Lino Grama, Ricardo M. Martins, Mauro Patrão et al.

    Same author Llohann Dallagnol Sperança

  6. On Mean curvature flow of Singular Riemannian foliations: Non compact cases

    Marcos M. Alexandrino, Leonardo F. Cavenaghi and Icaro GonçalvesWider flows

    Same author Leonardo Francisco Cavenaghi

  7. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows

    Shared terms “positive Ricci”

  8. Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

    Steven Gindi

    Topics Ricci solitons · Shared terms “structure group”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.