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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Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

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Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow on a closed (compact without boundary) manifold [Formula: see text]: [Formula: see text] In this paper, we prove the corresponding results on manifolds with boundary. More precisely, given a compact manifold [Formula: see text] with smooth boundary [Formula: see text], we study the convergence rate of the Yamabe flow with boundary: [Formula: see text] and the conformal mean curvature flow: [Formula: see text]