Paper · May 19, 2019

The Partial Ricci Flow on -foliations

Vladimir Rovenski, Robert Wolak

math.DGarXiv:1905.07704v4
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Abstract

In the paper we introduce new metric structures on -foliations that are less rigid than the well-known structures: almost contact and 3-quasi-Sasakian structures as well as -structures with parallelizable kernel and almost para--structures with complemented frames. We discuss the properties of the new structures in order to demonstrate similarities with the corresponding classical structures. Then using the flow of metrics on a -foliation, we build deformation retraction of our structures with positive partial Ricci curvature onto the subspace of the aforementioned classical structures.

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Submitted
May 19, 2019
Last revised
Jan 28, 2021 (v4)
Subjects
math.DG

Comments: 12 pages

Ricci flowSasakian & contact geometry

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  1. Characterization of Sasakian Manifolds by Curvature Conditions

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    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “partial Ricci”, “metric structure”, “Sasakian structure”

  2. Characterizations of weak almost -manifolds with curvature properties

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    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “partial Ricci”, “metric structure”, “almost contact”

  3. Generalized Ricci solitons and Einstein metrics on weak -contact manifolds

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  4. On non-gradient -quasi-Einstein contact metric manifolds

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    Topics Sasakian & contact geometry · Same author Vladimir Rovenski

  5. --Ricci solitons and Einstein metrics on a weak -Kenmotsu manifold

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    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “almost contact”

  6. -Ricci solitons and -Einstein metrics on weak -Kenmotsu -manifolds

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    Topics Sasakian & contact geometry · Same author Vladimir Rovenski · Shared terms “metric structure”, “foliation”

  7. --Ricci solitons on weak Kenmotsu -manifolds

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  8. Einstein-type metrics and generalized Ricci solitons on weak -K-contact manifolds

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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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sasakian & contact geometry Sasakian manifolds are odd-dimensional manifolds whose metric cone is Kähler. They carry a transverse Kähler structure, so Kähler–Ricci flow has an… Start with: THE SASAKI–RICCI FLOW (SMOCZYK et al., 2010)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.