*2.2. Coframe Bundle and Minkowski Bundle*

It is clear now that having *E* as a fiber bundle over *M* with fibers isomorphic to *F* and *F* ′ as a space equipped with the smooth action T ′ of *G*, implies the possibility of building a bundle *E* ′ *associated* to

<sup>6</sup> The bundle is said to be *trivial* if U = M.

*E*, which shares the same structure group and the same transition functions *gij*. By the fiber bundle construction theorem, we have a new bundle *E* ′ over *M* with fibers isomorphic to *F* ′ .

This bundle is called the *associated bundle* to *E*.

Depending on the nature of the associated bundle<sup>7</sup> , we have the following two definitions:

**Definition 4** (Associated *G*-principal bundle)**.** *Let π* : *E* → *M be a fiber bundle over a differentiable manifold M, G be a Lie group, F* ′ *be a topological space, and* P *be a smooth right action of G on F* ′ *. Let also E* ′ *be the associated bundle to E with fibers isomorphic to F*′ *.*

*If F* ′ *is the principal homogeneous space*<sup>8</sup> *for* P*, namely* P *acts freely and transitively on F* ′ *, then E* ′ *is called the G-principal bundle associated to E.*

**Definition 5** (Associated bundle to a *G*-principal bundle)**.** *Let P be a G-principal bundle over M, F* ′ *be a space, and ρ: G*→ Diffeo(*F* ′ ) *be a smooth effective left action of the group G on F*′ *.*

*We then have an induced right action of the group G over P* × *F* ′ *given by*

$$(p, f') \* \mathcal{g} = (\mathfrak{P}\_{\mathcal{S}}(p), \rho(\mathfrak{g}^{-1})(f')).\tag{7}$$

*We define the associated bundle E to the principal bundle P, as an equivalence relation:*

$$E := P \times\_{\rho} F' = \frac{P \times F'}{\sim} \, ' \tag{8}$$

*where* (*p*, *f* ′ ) ∼ (P*g*(*p*), *ρ*(*g* −1 )(*f* ′ ))*, p* ∈ *P, and f* ′ ∈ *F* ′ *with projection π<sup>ρ</sup>* : *E* → *M s.t. πρ*([*p*, *f* ′ ]) = *π*(*p*) = *x* ∈ *M.*

Therefore *π<sup>ρ</sup>* : *E* → *M* is a fiber bundle over *M* with *π* −1 *ρ* (*x*) ≃ *F* ′ for all *x* ∈ *M*.

**Observation 2:** The new bundle, given by the latter definition, is what we expected from a general associated bundle: a bundle with the same base space, different fibers, and the same structure group.

**Idea:** We take a *G*-principal bundle *P* as an associated bundle to *TM*, and we build a vector bundle associated to *P* with a fiber-wise metric *η*. We shall call this associated bundle V.

First of all, we display the *G*-principal bundle as the *G*-principal bundle associated to *TM*.

**Definition 6** (Orthonormal coframe)**.** *Let* (*M*, *g*) *be a pseudo-riemannian n-dimensional differentiable manifold and* (*V*, *η*) *be an n-dimensional vector space with minkowskian metric η. A coframe at x* ∈ *M is the linear isometry.*

$$\eta\_x e := \left\{ \,\_x e : \, T\_x M \to V \, \Big|\,\,\_x e^\* \eta := \eta\_{ab\,\,x} e^a \,\_x e^b = \mathrm{g} \right\},\tag{9}$$

*equivalently xe a forms an ordered orthonormal basis in T*∗ *<sup>x</sup> M. An orthonormal frame is defined as the dual of a coframe.*

<sup>7</sup> We will be dealing with two particular types of associated bundles: a principal bundle associated to a vector bundle and a vector bundle associated to a principal bundle.

<sup>8</sup> The space where the orbits of *G* span all the space.

**Observation 3:** Locally, coframes can be identified with local covector fields. A necessary and sufficient condition for identifying them with global covector fields (namely a coframe for each point of the manifold) is to have a *parallelizable* manifold, namely a trivial tangent bundle.

**Definition 7** (Orthonormal coframe bundle)**.** *Let* (*M*, *g*) *be a differentiable n-dimensional manifold with pseudo-riemannian metric g and T*∗*M be its cotangent bundle (real vector bundle of rank n).*

*We call the coframe bundle F* ∗ *O* (*M*) *the G-principal bundle where the fiber at x* ∈ *M is the set of all orthonormal coframes at x and where the group G* = O(*n* − 1, 1) *acts freely and transitively on them.*

The dual bundle of this is the orthonormal frame bundle, and it is denoted by *FO*(*M*), made up of orthonormal frames (dual of orthonormal coframes).

### **Observations 4:**


We are now ready to define tetrads.

**Definition 8** (Tetrads)**.** *Let ρ* : *O*(3, 1) → Aut(*V*) *be the fundamental representation.*

*Tetrads are the bundle isomorphisms <sup>e</sup>* : *TM* → V*. They are identifiable with elements <sup>e</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> (*M*, V)*, and if M is parallelizable, tetrads can be identified with* Ω<sup>1</sup> (*M*, *V*) ∋ *e <sup>a</sup>v<sup>a</sup> such that* {*va*} *is an orthonormal basis of V, e <sup>a</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> (*M*)*, and ηabe a e <sup>b</sup>* = *g.*

### **3. Principal Connection**
