  The St\"uckelberg analysis of nonlinear massive gravity in the presence of a
general fiducial metric is investigated. We develop a "covariant" formalism for
the St\"uckelberg expansion by working with a local inertial frame, through
which helicity modes can be characterized correctly. Within this covariant
approach, an extended $\Lambda_3$ decoupling limit analysis can be consistently
performed, which keeps $\bar{R}_{\mu\nu\rho\sigma}/m^2$ fixed with
$\bar{R}_{\mu\nu\rho\sigma}$ the Riemann tensor of the fiducial metric. In this
extended decoupling limit, the scalar mode $\pi$ acquires self-interactions due
to the presence of the curvature of the fiducial metric. However, the equation
of motion for $\pi$ remains of second order in derivatives, which extends the
understanding of the absence of the Boulware Deser ghost in the case of a flat
fiducial metric.
