A rigidity theorem for codimension one shrinking gradient Ricci solitons in $\mathbb R^{n+1}$

TL;DR

This paper proves a splitting theorem and a rigidity result for certain complete shrinking gradient Ricci solitons in Euclidean space, under nonnegative curvature conditions, advancing understanding of their geometric structure.

Contribution

It introduces a new rigidity theorem specifically for codimension one shrinking gradient Ricci solitons in Euclidean space with nonnegative Ricci curvature.

Findings

Established a splitting theorem for complete gradient Ricci solitons with nonnegative curvature.

Proved a rigidity theorem for codimension one shrinking gradient Ricci solitons in $\,\mathbb{R}^{n+1}$.

Enhanced understanding of the geometric structure of Ricci solitons under curvature conditions.

Abstract

We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.