In this article we study any 4-dimensional Riemannian manifold (M, g) with harmonic curvature which admits a smooth nonzero solution f to the following equationwhere Rc is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. We show that a neighborhood of any point in some open dense subset of M is locally isometric to one of the following five types;and H 2 (k) are the two-dimensional Riemannian manifold with constant sectional curvature k > 0 and k < 0, respectively, (iii) the static spaces in Example 3 below, (iv) conformally flat static spaces described in Kobayashi's [18], and (v) a Ricci flat metric.We then get a number of Corollaries, including the classification of the following four dimensional spaces with harmonic curvature; static spaces, Miao-Tam critical metrics and V -static spaces.The proof is based on the argument from a preceding study of gradient Ricci solitons [17]. Some Codazzi-tensor properties of Ricci tensor, which come from the harmonicity of curvature, are effectively used.where Rc is the Ricci curvature of g, x is a constant and y(R) a function of R. There are several well-known classes of spaces which admit such solutions.

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In this article we give a classification of three dimensional m‐quasi Einstein manifolds with two distinct Ricci‐eigen values. Our study provides explicit description of local and complete metrics and potential functions. We also describe the associated warped product Einstein manifolds in detail. For the proof we present a Codazzi tensor on any three dimensional m‐quasi Einstein manifold and use geometric properties of the tensor which help to analyze the m‐quasi Einstein equation effectively. A technical advance over preceding studies is made by resolving the case when the gradient ∇f of the potential function is not a Ricci‐eigen vector field.

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