*Is there any difference?*

In the ordinary formulation of General Relativity (as in the original Einstein's work, for instance), we have objects called <sup>Γ</sup>*s*, which are coefficients of a linear connection ∇ and thus determined by a parallel transport of tangent vectors.

The biggest advantage of treating O(3, 1) as an "explicit symmetry" of the theory is that we have obtained the possibility of defining a *principal connection*, which is the same kind of entity we have in an ordinary gauge theory<sup>9</sup> .

### *3.1. Ehresmann Connection*

If we consider a smooth fiber bundle *π* : *E* → *M*, where fibers are differentiable manifolds, we can of course take tangent spaces at points *e* ∈ *E*. Having the tangent bundle *TE*, we may wonder if it is possible to separate the contributions coming from *M* to the ones from the fibers.

This cannot be done just by stating *TE* = *TM* ⊕ *TF*, unless *E* = *M* × *F* is the trivial bundle. Namely, we cannot split directly vector fields on *M* from vector fields on the fibers *F*.

We can formalize this idea: use our projection *π* for constructing a tangent map *π*<sup>∗</sup> = *dπ* : *TE* → *TM*, and consider its kernel.
