On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

TL;DR

This paper investigates the fundamental group of complete steady gradient Ricci solitons with nonnegative sectional curvature, establishing conditions under which it is trivial or infinite, and characterizing their topological structure.

Contribution

It proves the fundamental group is either trivial or infinite and shows such solitons are diffeomorphic to Euclidean space under certain conditions.

Findings

Fundamental group is trivial or infinite.

Complete $ abla$-noncollapsed solitons with nonnegative curvature are diffeomorphic to $ r^n$.

Provides topological classification of these Ricci solitons.

Abstract

In this paper, we study the fundamental group of the complete steady gradient Ricci soliton with nonnegative sectional curvature. We prove that the fundamental group of such a Ricci soliton is either trivial or infinite. As a corollary, we show that an $n$-dimensional complete $\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature must be diffeomorphic to $\mathbb{R}^n$.