Within the standard Lagrangian settings (i.e., the difference between kinetic and potential energies), we discuss and report isochronicity, linearizability and exact solvability of some n-dimensional nonlinear position-dependent mass (PDM) oscillators. In the process, negative the gradient of the PDM-potential force field is shown to be no longer related to the time derivative of the canonical momentum, p = m (r) ṙ, but it is rather related to the time derivative of the pseudo-momentum, π (r) = m (r)ṙ (i.e., Noether momentum). Moreover, using some point transformation recipe, we show that the linearizability of the n-dimensional nonlinear PDM-oscillators is only possible for n = 1 but not for n ≥ 2. The Euler-Lagrange invariance falls short/incomplete for n ≥ 2 under PDM settings. An alternative invariance is therefore sought in the so called, hereinafter, n-dimensional PDM Ḣ-invariance (i.e., time derivative of the Hamiltonian). Such invariance, in addition to Newtonian invariance of Mustafa [42] authorizes, in effect, the use of the exact solutions of one system to find the solutions of the other. A sample of isochronous n-dimensional nonlinear PDM-oscillators examples are reported.