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We study harmonic metrics with respect to the class of invariant metrics on non-reductive homogeneous four dimensional manifolds. In particular, we consider harmonic lifted metrics with respect to the Sasaki lifts, horizontal lifts and complete lifts of the metrics under study.
The present paper is concerned with obtaining a classification regarding to four-dimensional semi-symmetric neutral Lie groups. Moreover, we discuss some geometric properties of these spaces. We exhibit a rich class of non-Einstein Ricci soliton examples.
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