Abstract
We study a non-trivial radially flat generalized m-quasi Einstein manifold M with finite m and Codazzi Ricci tensor, and obtain an explicit expression of the Ricci tensor over an open dense subset \(M^{*}\) of M on which the gradient of the potential function vanishes nowhere. Further, we prove that \(M^{*}\) is either Ricci-flat or is a non-steady m-quasi Einstein manifold which is locally a product of a line and an \((n-1)\)-dimensional Einstein space. In both the cases, \(M^{*}\) is conformally flat in dimension 4, and the Bach tensor vanishes in any dimension \(\ge 3\). Finally, we show that an m-quasi Einstein manifold with \(m \ne 1\) (finite) and constant scalar curvature admitting a non-parallel closed conformal vector field is Einstein.
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The authors would like to thank the referee for a valuable suggestion.
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The first author was funded by the University Grants Commission (UGC), India, in the form of Senior Research Fellowship.
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Poddar, R., Sharma, R. & Subramanian, B. Certain Classification Results on m-quasi Einstein Manifolds. Results Math 78, 197 (2023). https://doi.org/10.1007/s00025-023-01973-9
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DOI: https://doi.org/10.1007/s00025-023-01973-9
Keywords
- Generalized m-quasi Einstein manifold
- Codazzi tensor
- closed conformal vector field
- constant scalar curvature
- radially flat