The critical point equation on a four dimensional warped product manifold

摘要

On a compact oriented $n$-dimensional manifold $(M^n,$ $g)$, it has been conjectured that a metric $g$ satisfying the critical point equation (2) should be Einstein. In this paper, we prove that if a manifold $(M^4,g)$ is a $4$-dimensional oriented compact warped product, then $g$ can not be a solution of CPE with a non-zero solution function $f$.