Complete Ricci solitons on Finsler manifolds

TL;DR

This paper explores the properties of complete Ricci solitons on Finsler manifolds, showing that shrinking solitons have finite fundamental groups, thereby extending the understanding of geometric flows in Finsler geometry.

Contribution

It generalizes the concept of Ricci solitons to Finsler manifolds and proves that complete shrinking Ricci solitons have finite fundamental groups.

Findings

Complete shrinking Ricci solitons on Finsler manifolds have finite fundamental groups.

Provides a brief introduction to Finslerian Ricci flow and solitons.

Extends known results from Riemannian to Finsler geometry.

Abstract

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einstein metrics and are previously developed by the present authors for Finsler manifolds. In the present work, it is shown that a complete shrinking Ricci soliton Finsler manifold has a finite fundamental group.