The following procedure is described: Starting with a connection in a principal fiber bundle P(M,G), where G is either the Poincaré group or one of the de Sitter groups, a connection in the bundle of linear frames of a submanifold N of M is constructed by using the translational components of the original connection for frame identification. The dimension of N is gauge dependent, and the flat four-dimensional Minkowski space-time may appear of dimension less than 4 when certain gauges are used. It is shown that in the case of a de Sitter group, the minimum dimension to which the flat four-dimensional space-time can be reduced is 1, while the number is 0 for the Poincaré group. The gauge transformation that achieves the maximum dimension reduction in the de Sitter case is constant and leads to infinite strings as a result. Variable continuous gauge transformations that can reduce the dimension over a finite region of the base manifold are also considered.

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In order to explain the existence of internal quantum numbers of elementary particles, the ordinary space–time position vector is assumed to be a bilinear combination of two basic spinors. It is shown that this assumption leads to the replacement of the inhomogeneous Lorentz group by the (4 + 1) de Sitter group. Using the idea of external and internal space for the description of strongly interacting particles, the assumption provides a simple way of introducing the internal symmetries. The internal symmetry group SU(3) appears naturally connected with the external space, yielding the Gell-Mann -Okubo mass formula.