Our article is motivated by the vanishing condition of minimal volume of closed smooth manifolds $M$ that admit locally homogeneous Riemannian metric.We give a complete criterion of vanishing of $\norm{M}$ (a lower bound of $\Minvol(M)$, up to a multiple of dimension-dependent constant) for general $M$, and a complete criterion of the vanishing of $\Minvol(M)$ of closed homogeneous Riemannian manifolds. We also present a technical fibering lemma, which is useful throughout the article.