Frame bundle is parallelizable - Kobayashi Let $M$ be a Riemannian manifold, and let $L(M)$ be the associated frame bundle. At the end of page 40 of Kobayashi's book, as I understand, it is stated that:

There exsits $n^2$ connection forms $\omega_j^i$ on $L(M)$ which are nowhere vanishing. And hence together with the $n$ canonical/solder forms $\theta_i$, they give $L(M)$ an absolute parallelism. 

I could prove that the $n$ solder forms are nowhere vanishing. My question is, what are the Kobayashi's $n^2$ connection forms on $L(M)$? Are they related to the Levi-Civita connection forms on $M$? (which can be $0$ everywhere if I choose a flat manifold?). And how do we prove that they are nowhere vanishing (if it is not direct from the definition).
 A: A connection form $\omega$ on $L(M)$ is a $\mathfrak{gl}(n,\mathbb{R})$-valued 1-form on $L(M)$ (with $n=\dim M$) satisfying 
$$
(i)\quad \omega_p(p\cdot \xi) = \xi\quad \forall \xi\in\mathfrak{gl}(n,\mathbb{R}), p\in L(M) \\ 
(ii)\quad R_g^*\omega = \operatorname{Ad}_{g^{-1}}\omega \quad \forall g\in \operatorname{GL}(n,\mathbb{R}). \qquad
$$
The Lie algebra $\mathfrak{gl}(n,\mathbb{R})$ is spanned by the matrices $E^i_{\ j}$ with 1 in the $ij$th position, and zeros elsewhere, and so we can write $\omega = \sum_{i,j=1}^n\omega^i_{\ j}E^i_{\ j}$, where the $\omega^i_{\ j}$ are now $\mathbb{R}$-valued 1-forms on $L(M)$. Property (i) above ensures that they don't vanish on $L(M)$.
These forms are related to the Christoffel symbols $\Gamma^i_{kj}$ as follows: any coordinate patch $\phi:U\subset M\to\mathbb{R}^n$ induces a section $s_\phi:U\to L(M)$ by $s_{\phi}(x) = \left(\frac{\partial}{\partial x_1}\Big\vert_x,\ldots,\frac{\partial}{\partial x_n}\Big\vert_x\right)$. Then $s_\phi^*\omega^i_{\ j}$ will be a 1-form on $U\subset M$, and we can write $s_\phi^*\omega^i_{\ j} = \sum_{k=1}^n\Gamma^i_{kj}\,dx^k$ for some functions $\Gamma^i_{kj}:U\to\mathbb{R}$.
I would suggest you read Kobayashi and Nomizu - Foundations of Differential Geometry - Volume I for most of this material, especially Chapter III, Section 2.
