In this paper, we study the global nonexistence of solutions to a nonlinear wave equation with critical potential V ( x ) $V(x)$ on a Riemannian manifold, the form of which is more general than those in (Todorova and Yordanov in C. R. Acad. Sci., Sér. 1 Math. 300:557-562, 2000). The way we follow is motivated by the work of Qi S. Zhang (C. R. Acad. Sci., Sér. 1 Math. 333:109-114, 2001). We also prove the local existence and uniqueness result.
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