I'm pretty familiar with intrinsic geometry utilized in say General relativity for instance, and I understand the intrinsic curvature $\Omega$ 2-form of a connection $A$ on a manifold $M$ of dimension d.
$$\Omega=dA+A\wedge A$$
When trying to learn extrinsic geometry however, I find myself wanting. Consider some manifold isometrically $M$ embedded within another manifold $N$. Via the Gauss-Codazzi equation we get something like:
$$\Omega'=\Omega+\Pi\wedge\Pi$$
Where $\Pi$ is the second fundamental form. We can write the curvature on $M$ as:
$$\Omega=\Omega'-\Pi\wedge\Pi$$
I would like to better understand the meaning of the second fundamental form from the intrinsic point of view (yes, I know that sounds weird). Suppose the embedding I choose is within $M$'s own frame bundle (let's choose oriented orthonormal) $$M\subset F_{o}(M)$$. For some local trivialization, we have that $$F_{o}(M)=M\times SO(n)$$.
In the intrinsic case, we have curvature in terms of $A\subset SO(n)$, which would seem to be related to the second fundamental form for this choice of embedding.
What is the second fundamental form's relation to a G-connection when the embedding we choose is that principal G-bundle?