Geometric study of non-constant vector fields making hyperbolic space Ricci-Bourguignon solitons

TL;DR

This work analyzes vector fields on hyperbolic spaces $\mathbb{H}^n$ that induce Ricci-Bourguignon solitons, building on the result that such fields are Killing with a fixed algebraic form. It provides an explicit Lie-algebraic classification and flows for $n=2$, $n=3$, and general $n\ge 3$, identifying the generating sets (translations, expansion, and boosts/rotations) and a determinant-based criterion for when the dual form is a contact form in odd dimensions. It further derives exact expressions for the flows and computes the dual forms, showing preservation of the dual form by the soliton vector fields via $\mathcal{L}_X w=0$, while the dual form is not closed in these nontrivial cases. By connecting the algebraic structure $\Gamma_n\subset\mathfrak{so}(n,1)$ to geometric properties of RB-solitons, the results contribute to the understanding of self-similar solutions of the RB-flow in hyperbolic settings.

Abstract

The objective of this paper is to deepen the study of vector fields on hyperbolic spaces $\mathbb{H}^n$ that transform them into a Ricci-Bourguignon soliton. Starting from a recent work in \cite{bousso2025ricci} which characterizes these fields as Killing fields of a specific shape, we propose a detailed geometric study of their structure and behavior in the dimensions $n=2, 3$ and $n\geq 3$. In odd dimension we show that the dual form of these vectors are contact forms. This work aims to enrich the understanding of self-similar solutions of the Ricci flow in a more general context.

Theorems & Definitions (33)

• Theorem 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Theorem 3.5
• proof
• Proposition 3.6