Why a principal $\mathbb{R}$-bundle is always trivial?
I know that a principal bundle of the form $(E,B,\pi,G)$ is trivial if and only if it admits a global section $f:B\to E$. So which section should I take? Is there a simpler way to prove this property?
On
I think the following is a more rigorous version of what Tsemo Aristide was going for.
General setup
Fix a Lie group $G$ and a manifold $F$ on which $G$ acts simply transitively. Then, for a manifold $M$, let us denote by $\mathrm{Prin}_{F,G}(M)$ the isomorphism classes of principal $F$-bundles $T\to M$ with structure group $G$. In other words, such an object is
Then, if we let $\mathcal{O}_{M,G}$ be sheaf on $M$ given by
$$\mathcal{O}_{M,G}(U):=\mathrm{Hom}_\text{smooth}(U,G)$$
(where we multiply two maps by $$(f_f f_2)(u):=f_1(u)f_2(u)$$ which makes sense since $G$ is a group), then there is a bijection
$$\mathrm{Prin}_{F,G}(M)\cong \check{H}^1(M,\mathcal{O}_{M,G})$$
where this right-hand side is the first Cech cohomology. For the definition of this cohomology set see [Wedhorn, §7.2], and for this bijection see [Wedhorn, Equation (8.8)].
$\mathbb{R}$-bundles
To apply this setup to study $\mathbb{R}$-bundles, we observe that the set of isomorphism classes of $\mathbb{R}$-bundles on $M$ is precisely $\mathrm{Prin}_{\mathbb{R},\mathbb{R}}(M)$ where $G=F=\mathbb{R}$ and $G=\mathbb{R}$ acts on $F=\mathbb{R}$ by left translation.
Claim: Every $\mathbb{R}$-bundle on $M$ is isomorphic to the trivial $\mathbb{R}$-bundle $\mathbb{R}\times M$.
To prove this, we must show that $\check{H}^1(M,\mathcal{O}_{M,\mathbb{R}})$ is trivial. Note though that
$$\mathcal{O}_{M,\mathbb{R}}(U)=\mathrm{Hom}_\text{smooth}(U,\mathbb{R})$$
is nothing but the sheaf of smooth functions on $M$ which, for simplicitly, I just denote by $\mathcal{C}^\infty_M$. Now, the point here is then the following lemma which uses the existence of partitions of unity:
Lemma ([Wedhorn, Theorem 9.11]): Let $M$ be a smooth manifold. Then, $\mathcal{C}^\infty_M$ is a soft sheaf.
I won't define what this means exactly here (see [Wedhorn, §9.1] for details), but in essence it means that every section of $\mathcal{C}^\infty_M$ over a closed set $Z$ can be extended to a smooth function on all of $M$. You can see how proving such a thing would use bump functions, which in turn use the existence of partitions of unity (see [Wedhorn, Lemma 9.10 and Theorem 9.11] to have this fleshed out).
There is then the following general, but powerful fact from the theory of cohomology of sheaves.
Proposition ([Wedhorn, Theorem 9.14]): For a paracompact Hausdorff space $X$, and any soft sheaf of groups $\mathcal{G}$, the set $\check{H}^1(X,\mathcal{G})$ is a singleton.
The idea of the proof of this result is to refine an open cover $\{U_i\}$ that trivializes any $\mathcal{G}-torsor$ $\mathcal{F}$ (a generalization of a principal bundle) by a closed cover $\{Z_j\}$. Over each $Z_j$ one has a global section of $\mathcal{F}$, and using the softness property one can globalize these sections to a global section of $\mathcal{F}$, which implies its trivial.
We then easily get the following.
Proof of claim: This just comes from combining the above:
$$\{\mathbb{R}\text{-bundles on M}\}/\text{iso.}\cong \mathrm{Prin}_{\mathbb{R},\mathbb{R}}(M)\cong \check{H}^1(M,\mathcal{C}^\infty_M)=\{\ast\}$$
where the last equality follows since $\mathcal{C}^\infty_M$ is smooth. $\blacksquare$
References:
[Wedhorn] Wedhorn, T., 2016. Manifolds, sheaves, and cohomology. Springer Spektrum.
Since it is a differential geometry question suppose that $B$ is paracompact, let $(U_i)_{i\in I}$ a trivialisation, and $f_i$ a partition of unity subordinate to $(U_i)$. Let $h_i:U_1\rightarrow\mathbb{R}$ defined by $h_i(x)=1$, define $h=\sum_i f_ih_i$ since $f_i\geq 0, h_i>0$.