Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor

Huai-Dong Cao, Jiangtao Yu

Abstract

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat gradient steady Ricci solitons. More precisely, we prove that any $n$-dimensional complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well.

On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor

Abstract

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify -dimensional () complete -flat gradient steady Ricci solitons. More precisely, we prove that any -dimensional complete noncompact gradient steady Ricci soliton with vanishing -tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well.

Paper Structure

This paper contains 6 sections, 11 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical conformal tensors
  4. Two basic facts for gradient steady Ricci solitons
  5. The covariant 3-tensor $D_{ijk}$
  6. The proof of Theorem 1.1

Key Result

Theorem 1.1

Let $(M^n, g_{ij}, F)$, $n\ge 5$, be a complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor. Then $(M^n, g_{ij}, F)$ is either Ricci-flat with a constant potential function, or a quotient of the product steady soliton $N^{n-1} \times {\mathbb R}$, where $N^{n-1}$ is Ricci-flat

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.2
  • Proposition 2.1
  • ...and 8 more