Static and related critical spaces with harmonic curvature and three Ricci eigenvalues

J. Korean Math. Soc. Published online March 27, 2020

Jongsu Kim
Sogang University

Abstract : In this article we study about an $n$-dimensional Riemannian manifold $(M,g)$ with harmonic curvature and less than four Ricci eigenvalues which admits a smooth non constant solution $f$ to the following equation
\begin{eqnarray} \label{0002bxu}
\nabla df = f(r -\frac{R}{n-1} g) + x \cdot r+ y(R) g,
\end{eqnarray}
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 $R$.

We show that if $(M,g, f)$ with harmonic curvature and less than four Ricci eigenvalues satisfies (\ref{0002bx}), then in a neighborhood $V$ of each point in some open dense subset of $M$, one of the following three 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 metric of an interval and an Einstein manifold.

{\rm (iii)} $(V, g)$ is Einstein and isometric to a domain in the warped product of the form $g=ds^2 + (\frac{df}{ds})^2 \tilde{g}$, where $s$ is a function such that $\nabla s = \frac{ \nabla f }{|\nabla f|}$ and $ \tilde{g}$ is Einstein.

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For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.