  Motivated by the puzzling report of the observation of a ferromagnetic
insulating state in LaMnO$_3$/SrTiO$_3$ heterostructures, we calculate the
electronic and magnetic state of LaMnO$_3$, coherently matched to a SrTiO$_3$
square substrate within a "strained-bulk" geometry. We employ three different
density functional theory based computational approaches: (a) density
functional theory (DFT) supplemented with Hubbard U (DFT+U), (b) DFT +
dynamical mean field theory (DMFT), and (c) a hybrid functional treatment of
the exchange-correlation functional. While the first two approaches include
local correlations and exchange at Mn sites, treated in a static and dynamic
manner, respectively, the last one takes into account the effect of non-local
exchange at all sites. We find in all three approaches that the compressive
strain induced by the square substrate of SrTiO$_3$ turns LaMnO$_3$ from an
antiferromagnet with sizable orbital polarization to a ferromagnet with
suppressed Jahn-Teller distortion in agreement with experiment. However, while
both DFT+U and DFT+DMFT provide a metallic solution, only the hybrid
calculations result in an insulating solution, as observed in experiment. This
insulating behavior is found to originate from an electronic charge
disproportionation. Our conclusions remain valid when we investigate
LaMnO$_3$/SrTiO$_3$ within the experimental set-up of a superlattice geometry
using DFT+U and hybrid calculations.
