Direct proofs for spaces with cyclic parallel Ricci tensor
Abstract
We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial Q-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao--Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.
Related
- Non-Wellfounded and Cyclic Proofs for LTL: A Syntactic Correspondence with Linear Nested Sequents
- Two new proofs of Chui's Conjecture in Weighted Bergman Spaces
- Cyclic group tensors and the volume ratio of projective tensor products
- Common Fixed Points of Co-Cyclic and Sub-Cyclic Mappings in Complete Metric Spaces
- On the Motivic K-Theories of Classifying Spaces of Cyclic 2-Groups
BibTeX
@misc{zhang2026direct,
title = {{Direct proofs for spaces with cyclic parallel Ricci tensor}},
author = {Fengyuan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12930},
eprint = {2610.12930},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12930},
}
RIS
TY - UNPB
TI - Direct proofs for spaces with cyclic parallel Ricci tensor
AU - Fengyuan Zhang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12930
UR - https://arxiv.org/abs/2610.12930
AB - We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial Q-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao--Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.
ER -