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arXiv · 2610.12930

Direct proofs for spaces with cyclic parallel Ricci tensor

Fengyuan Zhang
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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.

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Fengyuan Zhang. 2026-10-09. Direct proofs for spaces with cyclic parallel Ricci tensor. https://arxiv.org/abs/2610.12930

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