*2.1. G-Principal Bundle*

We give some definitions<sup>2</sup> .

**Definition 1** (G-principal bundle<sup>3</sup> )**.** *Let M be a differentiable manifold and G be a Lie group.*

*A G-principal bundle P is a fiber bundle π* : *P* → *M together with a smooth (at least continuous) right action* <sup>P</sup> : *<sup>G</sup>* <sup>×</sup> *<sup>P</sup>* <sup>→</sup> *<sup>P</sup> such that* <sup>P</sup> *acts freely and transitively on the fibers*<sup>4</sup> *of P and such that π*(P*g*(*p*)) = *π*(*p*) *for all g* ∈ *G and p* ∈ *P.*

We need to introduce a fundamental feature of fiber bundles.

**Definition 2** (Local trivialization of a fiber bundle)**.** *Let E be a fiber bundle over M, a differentiable manifold, with fiber projection <sup>π</sup>* : *<sup>E</sup>* <sup>→</sup> *M, and let F be a space*<sup>5</sup> *.*

*A local trivialization* (*U*, *ϕU*) *of E, is a neighborhood U* ⊂ *M of u* ∈ *M together with a local diffeomorphism.*

$$
\mathfrak{gl}\_{\mathcal{U}} \colon \mathcal{U} \times \mathcal{F} \to \pi^{-1}(\mathcal{U}) \tag{1}
$$

*such that π*(*ϕU*(*u*, *f*)) = *u* ∈ *U for all u* ∈ *U and f* ∈ *F.*

This definition implies *π* −1 (*u*) ≃ *F* ∀*u* ∈ *U*.

**Definition 3** (Local trivialization of a *G*-principal bundle)**.** *Let P be a G-principal bundle.*

*A local trivialization* (*U*, *ϕU*) *of P is a neighborhood U* ⊂ *M of u* ∈ *M together with a local diffeomorphism.*

$$
\varphi\_{\mathcal{U}} : \mathcal{U} \times \mathcal{G} \to \pi^{-1}(\mathcal{U}) \tag{2}
$$

*such that π*(*ϕU*(*u*, *g*)) = *u* ∈ *U for all u* ∈ *U and g* ∈ *G and such that*

$$
\mathfrak{g}\_{\mathcal{U}}^{-1}(\mathfrak{P}\_{\mathcal{S}}(p)) = \mathfrak{g}\_{\mathcal{U}}^{-1}(p)\mathfrak{g} = (\mathfrak{u}\_{\prime}\mathfrak{g}^{\prime})\mathfrak{g} = (\mathfrak{u}\_{\prime}\mathfrak{g}^{\prime}\mathfrak{g}).\tag{3}
$$

<sup>1</sup> Disjoint union of tangent spaces: *TM* <sup>=</sup> <sup>∪</sup>*x*∈*M*{*x*} × *<sup>T</sup>x<sup>M</sup>*

<sup>2</sup> References [2–4] are recommended for further details.

<sup>3</sup> We give the definition based on our purposes; in general, we can release some hypotheses. In particular, *G* needs to be only a locally compact topological group and *M* needs to be a topological Hausdorff space. This definition is a version with a stronger hypothesis than the one contained in Reference [5].

<sup>4</sup> Fibers are *π* −1 (*x*) ∀*x* ∈ *M*.

<sup>5</sup> In the present case, *F* will be a differentiable manifold, a vector space, a topological space, or a topological group. Furthermore, if we write "space", we mean one among these.

**Observation 1:** A fiber bundle is said to be *locally trivial* in the sense that it admits a local trivialization for all *x* ∈ *M*, namely there exists an open cover {*Ui*} of *M* and a set of diffeomorphisms *ϕ<sup>i</sup>* such that every {*U<sup>i</sup>* , *<sup>ϕ</sup>i*} is a local trivialization<sup>6</sup> .

Here, we recall the similarity with a differentiable manifold. For a manifold when we change charts, we have an induced diffeomorphism between the neighborhoods of the two charts, given by the composition of the two maps.

Thus, having two charts (*U<sup>i</sup>* , *φi*) and (*U<sup>j</sup>* , *φj*), we define the following:

$$
\phi\_{\dot{j}} \circ \phi\_{\dot{i}}^{-1} : \phi\_{\dot{i}}(\mathcal{U}\_{\dot{i}} \cap \mathcal{U}\_{\dot{j}}) \to \phi\_{\dot{j}}(\mathcal{U}\_{\dot{i}} \cap \mathcal{U}\_{\dot{j}}).\tag{4}$$

At a level up, we have an analogous thing when we change trivialization. Of course, here, we have one more element: the element of fiber.

Taking two local trivializations (*U<sup>i</sup>* , *ϕi*) and (*U<sup>j</sup>* , *ϕj*) and given a smooth left action T : *G* → Diffeo(*F*) of *G* on *F*, we then have

$$((\varrho\_{\dot{j}}^{-1} \circ \varrho\_{\dot{i}})(\mathbf{x}, f) = (\mathbf{x}, \mathcal{T}(\varrho\_{\dot{i}\dot{j}}(\mathbf{x}))(f)) \qquad \forall \mathbf{x} \in \mathcal{U}\_{\dot{l}} \cap \mathcal{U}\_{\dot{l}\iota}f \in \mathcal{F}. \tag{5}$$

where the maps *gij* : *U<sup>i</sup>* ∩ *U<sup>j</sup>* → *G* are called the *transition functions* for this change of trivialization and *G* is called the structure group.

Such functions obey the following transition functions conditions for all *x* ∈ *U<sup>i</sup>* ∩ *U<sup>j</sup>* :


The last condition is called the *cocycle condition*.

**Theorem 1** (Fiber bundle construction theorem)**.** *Let M be a differentiable manifold, F be a space, and G be a Lie group with faithful smooth left action* T : *G* → Diffeo(*F*) *of G on F.*

*Given an open cover* {*Ui*} *of M and a set of smooth maps,*

$$\{t\_{ij} : \mathcal{U}\_i \cap \mathcal{U}\_j \to \mathcal{G}\} \tag{6}$$

*defined on each nonempty overlap, satisfying the transition function conditions.*

*Then, there exists a fiber bundle π* : *E* → *M such that*


A proof of the theorem can be found in Reference [6] (Chapter 1).
