Standard horizontal vector fields on the bundle of linear frames

Question

I'm reading Kobayashi & Nomizu Fondation of differential geometry and I got stuck in understanding what makes the standard horizontal vectors $B_{k}$ defined globaly on $P$ where $P$ is the bundle of linear frames? in the following proposition from kobayashi P.122 .

Answer 1

Let $M$ be an $n$-dimensional manifold, the bundle of linear frames $p:P\rightarrow M$ is the bundle such that for every $x\in M$, $P_x$ the fibre of $x$ is $\{u:T_xM\rightarrow \mathbb{R}^n\}$. This enables to define the fundamental form $\theta:TP\rightarrow \mathbb{R}^n$ such that for every $y\in P$ with $p(y)=x$ (remark that $y$ is linear map $y:T_xM\rightarrow\mathbb{R}^n$), $v\in T_yP$, $\theta_y(v)=y(dp_y(v))$. where $dp$ is the differential of the bundle map.

Let $\omega$ be a connection. It is defined by a distribution $H$ supplementary to the fibre of $p$, therefore the restriction of $\theta_y$ to $H_y$ is bijective. For every $u\in \mathbb{R}^n$ one can define $(\theta_y)_{\mid H_y}^{-1}(u)$ which is differentiable and globally defined.