 In the supplement  A Modern Formulation of Riemann's Theory  we touch on the idea of a fiber bundle, formed by two smooth manifolds F and M, bound together by a “projection” mapping π of F onto M, which partitions the manifold F into “fibers”, mapped by π to the different points of M. A fiber bundle &lt;F,M,π&gt; becomes a principal fiber bundle &lt;F,M,π,G&gt; if a Lie group G, known as the bundle's structure group, acts on F in such a way that each fiber of F is an orbit of the action and a few other conditions are met. For instance, the Lorentz group is the structure group of the principal fiber bundle of tetrads (orthonormal 4-tuples of tangent vectors at each point) on any relativistic spacetime, no matter how bizarre. In such ways, Lie groups provide a means of unifying the many models allowed by a physical theory and of introducing some degree of homogeneity among them. 