Journal of the
Korean Mathematical Society JKMS

In this article we make a local classification of $n$-dimensional Riemannian manifolds $(M,g)$ with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution $f$ to the following equation \begin{align} \label{0002bxu} \nabla df = f(r -\frac{R}{n-1} g) + x \cdot r+ y(R) g, \end{align} where $\nabla $ is the Levi-Civita connection of $g$, $r$ is the Ricci tensor of $g$, $x$ is a constant and $y(R)$ a function of the scalar curvature $R$. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, either {\rm (i)} or {\rm (ii)} below holds; {\rm (i)} $(V, g, f+x)$ is a static space and isometric to a domain in the Riemannian product of an Einstein manifold $N$ and a static space $(W, g_W, f+x)$, where $g_W$ is a warped product metric of an interval and an Einstein manifold. {\rm (ii)} $(V, g)$ is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

Keywords: Static space, critical metric, harmonic curvature

MSC numbers: 53C21, 53C25

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B4004460)