The standard definition of space-time perturbation is reexamined. It is seen that the noninvariance of the metric under identification gauge transformations is a consequence of the adopted zero signature in the fifth dimension of the space of space-times. An n-parameter extension of that definition is proposed, with a (4+n)-dimensional flat space of space-times with a nonsingular metric. It is shown that in the vicinity of a point in the background space-time there is a geometrically defined family of perturbations, which are solutions of the Einstein–Yang–Mills equations.