  The presence of a constant background antisymmetric tensor for open strings
or D-branes forces the space-time coordinates to be noncommutative. An
immediate consequence of this is that all fields get complexified. By applying
this idea to gravity one discovers that the metric becomes complex. Complex
gravity is constructed by gauging the symmetry $U(1,D-1)$. The resulting action
gives one specific form of nonsymmetric gravity. In contrast to other theories
of nonsymmetric gravity the action is both unique and gauge invariant. It is
argued that for this theory to be consistent one must prove the existence of
generalized diffeomorphism invariance. The results are easily generalized to
noncommutative spaces.
