The Generalized Ricci Flow for 3D Manifolds with One Killing Vector

TL;DR

This paper extends the Ricci flow equations to three-dimensional manifolds with a Killing vector by incorporating gauge fields and De Turck terms, resulting in a reduced system that admits exact solutions describing homogeneous geometries.

Contribution

It introduces a generalized Ricci flow for 3D manifolds with a Killing vector, including gauge fields and De Turck modifications, extending previous 2D results.

Findings

Flow reduces to Toda and linearized Toda equations.

Exact solutions describe homogeneous, anisotropic geometries.

Flow converges to specific geometric structures.

Abstract

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riemannian metric with one Killing vector. In particular, we show that the RG flow with De Turck terms can be reduced to two equations: the continual Toda flow solved by Bakas, plus its linearizaton. We find exact solutions which flow to homogeneous but not always isotropic geometries.