As on arXiv
Abstract
The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \beginequation D_ijk=\nabla_iR_jk+\nabla_jR_ki+\nabla_kR_ij=0\nonumber \endequation These structures include non-compact gradient shrinking Ricci soliton, compact -quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.
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Direct proofs for spaces with cyclic parallel Ricci tensor
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Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.