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We study the geometrical meaning of higher-order terms in matrix models ofYang-Mills type in the semi-classical limit, generalizing recent resultsarXiv:1003.4132 to the case of 4-dimensional space-time geometries with generalPoisson structure. Such terms are expected to arise e.g. upon quantization ofthe IKKT-type models. We identify terms which depend only on the intrinsicgeometry and curvature, including modified versions of the Einstein-Hilbertaction, as well as terms which depend on the extrinsic curvature. Furthermore,a mechanism is found which implies that the effective metric G on thespace-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.

Curvature and Gravity Actions for Matrix Models II: the case of general Poisson structure