  We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted
Riemannian manifolds and apply it to obtain a pinching theorem for closed
$f$-minimal hypersurfaces immersed in the product manifold
$\mathbb{S}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with $f=\frac {t^2}{4}$. Also we
classify closed $f$-minimal hypersurfaces with $L_f$-index one immersed in
$\mathbb{S}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with the same $f$ as above.
