  We study the geometrical meaning of higher-order terms in matrix models of
Yang-Mills type in the semi-classical limit, generalizing recent results
arXiv:1003.4132 to the case of 4-dimensional space-time geometries with general
Poisson structure. Such terms are expected to arise e.g. upon quantization of
the IKKT-type models. We identify terms which depend only on the intrinsic
geometry and curvature, including modified versions of the Einstein-Hilbert
action, as well as terms which depend on the extrinsic curvature. Furthermore,
a mechanism is found which implies that the effective metric G on the
space-time brane M \subset R^D "almost" coincides with the induced metric g.
Deviations from G=g are suppressed, and characterized by the would-be U(1)
gauge field.
