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Ricci Solitons on the Poincaré upper half plane

Abdou Bousso, Ameth Ndiaye

TL;DR

The work addresses Ricci solitons on the Poincaré upper half-plane by deriving and solving the soliton equations on $\mathbb{H}^2$ and then generalizing to $\mathbb{H}^n$. It delivers explicit forms for the soliton vector fields, analyzes when solutions are gradient or Killing, and reveals harmonicity properties of the components in low dimensions while highlighting nonharmonicity in higher dimensions. The results illuminate how soliton structures behave on negatively curved spaces and provide concrete expressions for geodesic-flow-related quantities. Overall, the paper advances the understanding of Ricci solitons and their generalizations in hyperbolic geometry with potential applications to geometric flows on non-Euclidean manifolds.

Abstract

In this paper, we characterize the Ricci soliton equations on the Poincaré upper half plane . First we classify all Ricci soliton and Ricci Bourguignon soliton in the half plane of Poincaré and after we generalize those equations in $\mathbb{H}^n$. We obtain some nice properties of the soliton about their geodesic flows.

Ricci Solitons on the Poincaré upper half plane

TL;DR

The work addresses Ricci solitons on the Poincaré upper half-plane by deriving and solving the soliton equations on and then generalizing to . It delivers explicit forms for the soliton vector fields, analyzes when solutions are gradient or Killing, and reveals harmonicity properties of the components in low dimensions while highlighting nonharmonicity in higher dimensions. The results illuminate how soliton structures behave on negatively curved spaces and provide concrete expressions for geodesic-flow-related quantities. Overall, the paper advances the understanding of Ricci solitons and their generalizations in hyperbolic geometry with potential applications to geometric flows on non-Euclidean manifolds.

Abstract

In this paper, we characterize the Ricci soliton equations on the Poincaré upper half plane . First we classify all Ricci soliton and Ricci Bourguignon soliton in the half plane of Poincaré and after we generalize those equations in . We obtain some nice properties of the soliton about their geodesic flows.
Paper Structure (7 sections, 13 theorems, 95 equations)

This paper contains 7 sections, 13 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ricci solitons
  4. Geometric of the Poincaré upper half plane
  5. Main results
  6. Poincaré half plane of dimension 2
  7. Poincaré upper half plane $\mathbb{H}^n$

Key Result

Proposition 2.5

Oneil A Riemannian manifold $(M^n, g)$ has constant curvature $k$ if and only if $Ric = (n-1)kg$.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 15 more