Compact Einstein warped product spaces with nonpositive scalar curvature

By Dong-Soo Kim, Young Ho Kim. 2003

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A remark on Einstein warped products (2010, background): “Indeed, in [3], elaborating on [7], it is observed the following characterization of quasi-Einstein metrics.”
On Bach flat warped product Einstein manifolds (2011, result): “If m is positive, then from [Kim and Kim 2003] and the comparison theorem of m-Bakry–Emery tensors in [Qian 1997, Theorem 5], a (λ, n+m)-Einstein manifold is compact if and only if λ > 0.”
On the geometry of gradient einstein-type manifolds (2014, background): “Other examples are, for instance, Ricci almost solitons ([41]), Yamabe solitons ([19], [31],[21], [9]), Yamabe quasi-solitons ([27], [45]), conformal gradient solitons ([43], [15]), quasi-Einstein manifolds ([30], [12], [16], [26], [35]), ρ-Einstein solitons ([17], [18]).”
On the classification of warped product Einstein metrics (2010, methodology): “As mentioned above, when ∂M = ∅, (λ, n+m)-Einstein metrics have been studied in [17, 30].”
Rigidity of Quasi-Einstein Metrics (2008, background): “The same result is proven in [10] for quasi-Einstein metrics on compact manifolds with finite m.”