Bulletin of the
Korean Mathematical Society BKMS

Consider the $L^2$-adjoint $s_g'^*$ of the linearization of the scalar curvature $s_g$. If $\ker s_g'^*\neq 0$ on an $n$-dimensional compact manifold, it is well known that the scalar curvature $s_g$ is a non-negative constant. In this paper, we study the structure of the level set $\varphi^{-1}(0)$ and find the behavior of Ricci tensor when $\ker s_g'^*\neq 0$ with $s_g>0$. Also for a non-trivial solution $(g,f)$ of $z=s_g'^*(f)$ on an $n$-dimensional compact manifold, we analyze the structure of the regular level set $f^{-1}(-1)$. These results give a good understanding of the given manifolds.

Keywords: scalar curvature, the regular level set, the traceless Ricci tensor

MSC numbers: Primary 53C25