Paper · Aug 9, 2026

Sasaki with torsion manifolds and string backgrounds

Beatrice Brienza, Anna Fino, Udhav Fowdar, Gueo Grantcharov

math.DGarXiv:2608.08781v2
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Abstract

Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a -Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact -Einstein manifold in dimension and . We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with .

Comments: 45 pages. Added new results on the flow for Sasaki with torsion structures

Ricci flowGeneralized & G₂ flowsEinstein metricsKähler geometryBoundaries & non-compactSasakian & contact geometry

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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 2021, 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 Geometric Flows of G₂-Structures on 3-Sasakian 7-Manifolds (Aaron Kennon and Jason D. Lotay, 2022)6Kennon 2022 Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds (Vladimir Rovenski, 2023)4Rovenski 2023 The weighted Hermite–Einstein equation (Michael Hallam and Abdellah Lahdili, 2024)8Hallam 2024 Some remarks on strong G₂-structures with torsion (Anna Fino and Udhav Fowdar, 2025)1Fino 2025 The canonical symmetry reduction of string backgrounds (Aaron Kennon and Jeffrey Streets, 2025)2Kennon 2025 Hermitian curvature flow and HKT geometry (Beatrice Brienza and Jeffrey Streets, 2026)5Brienza 2026 Topology of low-dimensional generalized Ricci solitons and string backgrounds (Jeffrey Streets, 2026)7Streets 2026 Almost Hermitian geometry and generalized Ricci flow (Anna Fino and Julieth Saavedra, 2026)3Fino 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some remarks on strong -structures with torsion

    Anna Fino and Udhav Fowdar PDF

    Topics Generalized & G₂ flows · Same authors Anna Fino, Udhav Fowdar · Shared terms “pluriclosed flow”, “Hermitian”, “equivalent”

  2. The canonical symmetry reduction of string backgrounds

    Aaron Kennon and Jeffrey Streets PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows · Shared terms “string backgrounds”, “almost contact”, “Hermitian”

  3. Almost Hermitian geometry and generalized Ricci flow

    Anna Fino and Julieth Saavedra PDF

    Topics Generalized & G₂ flows · Same author Anna Fino · Shared terms “Hermitian geometry”, “Bismut”, “pluriclosed”

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

    Vladimir Rovenski PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Shared terms “metric structure”, “characterize”

  5. Hermitian curvature flow and HKT geometry

    Beatrice Brienza and Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Einstein metrics, Kähler geometry · Same author Beatrice Brienza · Shared terms “Hermitian”, “equivalent”, “preserves”

  6. Geometric Flows of -Structures on 3-Sasakian 7-Manifolds

    Aaron Kennon and Jason D. Lotay PDF

    Topics Sasakian & contact geometry, Generalized & G₂ flows, Einstein metrics · Shared terms “geometric flows”

  7. Topology of low-dimensional generalized Ricci solitons and string backgrounds

    Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Shared terms “string backgrounds”, “Bismut”, “Hermitian”

  8. The weighted Hermite–Einstein equation

    Michael Hallam and Abdellah Lahdili PDF

    Topics Sasakian & contact geometry, Einstein metrics, Kähler geometry · Shared terms “Hermite”, “Sasaki”, “analogue”

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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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

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  3. 11A.1
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    A heat equation for the metric: spheres, cylinders and scaling.