  In this paper, we prove a classification theorem for the stable compact
minimal submanifolds of the Riemannian product of an $m_1$-dimensional
($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian
manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies
$\frac{1}{\sqrt{m_1-1}}\leq K_{M_1}\leq 1.$ This gives a generalization to the
results of F. Torralbo and F. Urbano [9], where they obtained a classification
theorem for the stable minimal submanifolds of the Riemannian product of a
sphere and any Riemannian manifold. In particular, when the ambient space is an
$m$-dimensional ($m\geq3$) complete hypersurface $M$ in the Euclidean space, if
the sectional curvature $K_{M}$ of $M$ satisfies $\frac{1}{\sqrt{m+1}}\leq
K_{M}\leq 1$, then we conclude that there exist no stable compact minimal
submanifolds in $M$.
