Is any minimal hypersurface also a minimal submanifold?

Question

Recently I learned that a minimal submanifold is one whose mean curvature vector $\vec{H}$ vanishes identically, which makes me wonder whether a minimal hypersurface qualifies as a minimal submanifold because I was told that a minimal hypersurface is a hypersurface $M$ whose mean curvature (scalar) $H$ vanishes identically and I have shown $\langle\vec{H},N\rangle=H$ if $N$ denotes a unit normal vector field along $M$. From this expression, we can see clearly that if $\vec{H}$ vanishes everywhere, so does $H$. But how about the converse? Can we conclude that $\vec{H}\equiv\mathbf{0}$, if $H\equiv0$? Thank you.