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In this article, we study homogeneous warped product Einstein metrics and their connections with homogeneous Ricci solitons. We show that homogeneous (λ, n + m)-Einstein manifolds (which are the bases of homogeneous warped product Einstein metrics) are one-dimensional extensions of algebraic solitons. This answers a question from a paper of He, Petersen and Wylie, where they prove the converse statement. Our proof is strongly based on their results, but it also makes use of sharp tools from the theory of homogeneous Ricci solitons. As an application, we obtain that any homogeneous warped product Einstein metric with homogeneous base is diffeomorphic to a product of homogeneous Einstein manifolds.
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