  The effective geometry and the gravitational coupling of nonabelian gauge and
scalar fields on generic NC branes in Yang-Mills matrix models is determined.
Covariant field equations are derived from the basic matrix equations of
motions, known as Yang-Mills algebra. Remarkably, the equations of motion for
the Poisson structure and for the nonabelian gauge fields follow from a matrix
Noether theorem, and are therefore protected from quantum corrections. This
provides a transparent derivation and generalization of the effective action
governing the SU(n) gauge fields obtained in [1], including the would-be
topological term. In particular, the IKKT matrix model is capable of describing
4-dimensional NC space-times with a general effective metric. Metric
deformations of flat Moyal-Weyl space are briefly discussed.
