PreprintCC BY 4.0arXiv
On Generalized Quasi-Einstein Manifolds
Abstract
In this paper, we study generalized m-quasi-Einstein (Mn,g,X,λ) under natural conditions on the potential vector field. We show that, under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on m and λ, including triviality results. We also revisit a recent theorem of Ghosh \cite{ghosh}, discuss a subtle issue in the argument, and provide a new formulation and proof. Finally, we establish rigidity results for manifolds with geodesic potential vector fields.
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BibTeX
@misc{carvalho2026generalized,
title = {{On Generalized Quasi-Einstein Manifolds}},
author = {Alcides de Carvalho and Anderson Lima and W. O. Costa-Filho},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.05473},
eprint = {2605.05473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05473},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Generalized Quasi-Einstein Manifolds
AU - Alcides de Carvalho
AU - Anderson Lima
AU - W. O. Costa-Filho
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2605.05473
UR - https://arxiv.org/abs/2605.05473
AB - In this paper, we study generalized m-quasi-Einstein (M^n,g,X,λ) under natural conditions on the potential vector field. We show that, under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on m and λ, including triviality results. We also revisit a recent theorem of Ghosh \cite{ghosh}, discuss a subtle issue in the argument, and provide a new formulation and proof. Finally, we establish rigidity results for manifolds with geodesic potential vector fields.
ER -