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Consider the example of the relationship of the group O(3) of rotations in3-space to the special unitary group SU(2). Given other unitary groups, whattransformations can we find? In this paper we describe a method of constructingtransformations in an N-squared dimensional manifold based on the properties ofthe unitary group U(N). With N = 2, the method gives the familiar rotations,boosts, translations and scale transformations of four dimensional spacetime.In the language of spacetime now applied for any N, we show that rotationspreserve distances and times. There is a scale-changing transformation. For Nmore than 2, boosts do not preserve generalized spacetime intervals, but eachN-squared dimensional manifold contains a 3+1 dimensional subspace withinvariant spacetime intervals. That each manifold has a four dimensionalspacetime subspace correlates with the property of U(N) that U(2) is asubgroup.

U(N) Based Transformations in N-Squared Dimensions