We study any fourdimensional Riemannian manifold
$\left(M,g\right)$
with harmonic curvature which admits a smooth nonzero solution
$f$ to the
equation
$$\nabla \phantom{\rule{0.3em}{0ex}}df=f\left(Rc\frac{R}{n1}g\right)+xRc+y\left(R\right)g,$$
where
$Rc$ is the
Ricci tensor of
$g$,
$x$ is a constant and
$y\left(R\right)$ 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: (i)
${\mathbb{S}}^{2}\left(\frac{R}{6}\right)\times {\mathbb{S}}^{2}\left(\frac{R}{3}\right)$ with
$R>0$, (ii)
${\mathbb{H}}^{2}\left(\frac{R}{6}\right)\times {\mathbb{H}}^{2}\left(\frac{R}{3}\right)$ with
$R<0$,
where
${\mathbb{S}}^{2}\left(k\right)$
and
${\mathbb{H}}^{2}\left(k\right)$
are the twodimensional Riemannian manifolds with constant sectional curvatures
$k>0$ and
$k<0$,
respectively, (iii) the static spaces we describe in Example 3, (iv) conformally flat
static spaces described by Kobayashi (1982), and (v) a Ricci flat metric.
We then get a number of corollaries, including the classification of the following
fourdimensional spaces with harmonic curvature: static spaces, Miao–Tam critical metrics
and
$V$static
spaces.
For the proof we use some Codazzitensor properties of the Ricci tensor and
analyze the equation displayed above depending on the various cases of multiplicity
of the Riccieigenvalues.
