Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Ricci flow for circle bundles over surfaces

Arash Bazdar, Georgios Fotopoulos

TL;DR

This work analyzes the normalized Ricci flow on circle bundles $P\to M$ over surfaces equipped with locally homogeneous $S^1$-connections by deriving a bundle metric $g_A=\pi^*g+f^2\omega\otimes\omega$ and reducing the flow to a finite-dimensional ODE system. Under a constant-curvature base and a Yang–Mills connection, the flow simplifies to a two-parameter evolution with a fixed relation $f_t=\lambda_t^{-1}$, allowing explicit solutions and asymptotic analysis. The authors classify the resulting geometries arising in the flow, obtaining six model geometries from Thurston's list (including $S^2\times\mathbb{R}$, $\mathrm{Nil}$, and $S^3$) and detailing how the base and fiber scale evolve in time depending on $K_0$ and $F_0$. This approach provides a tractable framework to study Ricci flow on bundles by converting it to ODE dynamics and highlights potential extensions to higher-rank bundles such as $\mathrm{SU}(2)$-bundles, with implications for understanding geometric evolution on Seifert-fibered and related manifolds.

Abstract

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

The Ricci flow for circle bundles over surfaces

TL;DR

This work analyzes the normalized Ricci flow on circle bundles over surfaces equipped with locally homogeneous -connections by deriving a bundle metric and reducing the flow to a finite-dimensional ODE system. Under a constant-curvature base and a Yang–Mills connection, the flow simplifies to a two-parameter evolution with a fixed relation , allowing explicit solutions and asymptotic analysis. The authors classify the resulting geometries arising in the flow, obtaining six model geometries from Thurston's list (including , , and ) and detailing how the base and fiber scale evolve in time depending on and . This approach provides a tractable framework to study Ricci flow on bundles by converting it to ODE dynamics and highlights potential extensions to higher-rank bundles such as -bundles, with implications for understanding geometric evolution on Seifert-fibered and related manifolds.

Abstract

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

Paper Structure

This paper contains 10 sections, 4 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Locally homogeneous S1̂-connections over surfaces
  3. Metrics on principal K-bundles
  4. The Levi-Civita connection associated with a bundle metric on circle bundles over surfaces
  5. The curvature tensor of a bundle metric on S1̂-bundles over surfaces
  6. The Ricci curvature of a metric bundle on principal S1̂-bundles over surfaces
  7. The Ricci flow equation of a metric bundle on principal S1̂-bundles over surfaces
  8. The Ricci flow equation on principal S1̂-bundles over surfaces
  9. Solutions of the normalized Ricci flow equation for locally homogeneous S1̂-connections on surfaces
  10. Conclusion

Key Result

Proposition 2.2

Let $\pi:P\to M$ be a principal $\mathbb S^1$-bundle over a surface $(M,g)$ endowed with a connection $A$. Using the same notation as above, the Levi-Civita connection $(\theta^i_j)$ of the bundle metric $g_A$ is given by

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5