  With inspiration from the K\"ahler geometry, we introduce a metric structure
on the energy class, $\mathcal{E}_{1,m}$, of $m$-subharmonic functions with
bounded energy and show that it is complete. After studying how the metric
convergence relates to the accepted convergences in this
Caffarelli-Nirenberg-Spruck model, we end by constructing geodesics in a
subspace of our complete metric space.
