Four dimensional static and related critical spaces with harmonic curvature

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 equation \begin{eqnarray} \label{0002bx} \nabla df = f(Rc -\frac{R}{n-1} g) + x Rc+ y® g. \end{eqnarray} where $Rc$ 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; {\rm (i)} $ $\mathbb{S}$^2(\frac{R}{6}) \times $\mathbb{S}$^2(\frac{R}{3})$with$R>0$,(\mathrm{i}\mathrm{i})$ $\mathbb{H}$^2(\frac{R}{6}) \times $\mathbb{H}$^2(\frac{R}{3}) $with$R<0$,where$$\mathbb{S}$^2(k) $and$$\mathbb{H}$^2(k) $ are the two-dimensional Riemannian manifold with constant sectional curvature $k>0$ and $k<0$, respectively, {\rm (iii)} the static spaces in Example 3 below, {\rm (iv)} conformally flat static spaces described in Kobayashi’s \cite{Ko}, and {\rm (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 \cite{Ki}. Some Codazzi-tensor properties of Ricci tensor, which come from the harmonicity of curvature, are effectively used.