We present a novel geometric port-Hamiltonian formulation of redundant manipulators performing a differential kinematic task $η=J(q)\dot{q}$, where $q$ is a point on the configuration manifold, $η$ is a velocity-like task space variable, and $J(q)$ is a linear map representing the task, for example the classical analytic or geometric manipulator Jacobian matrix. The proposed model emerges from a change of coordinates from canonical Hamiltonian dynamics, and splits the standard Hamiltonian momentum variable into a task-space momentum variable and a null-space momentum variable. Properties of this model and relation to Lagrangian formulations present in the literature are highlighted. Finally, we apply the proposed model in an \textit{Interconnection and Damping Assignment Passivity-Based Control} (IDA-PBC) design to stabilize and shape the impedance of a 7-DOF Emika Panda robot in simulation.

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