  We find a Simons type formula for submanifolds with parallel mean curvature
vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where
$M^n(c)$ is a space form with constant sectional curvature $c$, and then we use
it to prove a gap theorem for the mean curvature of certain complete
proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces
in $\mathbb{S}^n(c)\times\mathbb{R}$.
