Geometry of submanifolds via moving frames

Consider, for simplicity, a 2-dimensional manifold $\Sigma$ embedded into the 3-dimensional Riemannian manifold $(M,g)$, in such a way that the given frame $\{e_1,e_2,e_3\}$ is adapted to $\Sigma$, i.e., $\omega^3|_{T\Sigma}=0$, where $\omega^1,\omega^2,\omega^3$ is the dual coframe.
The surface $\Sigma$ inherits a Riemannian metric from the ambient manifold, with its corresponding Levi-Civita connection. We will denote by $\tilde{\omega}^1,\tilde{\omega}^2$ the restrictions of $\omega^1,\omega^2$ to $T\Sigma$, and by $\tilde{\Theta}$ and $\tilde{\Omega}$ the connection forms and the curvature forms, respectively, of the inherited connection.

According to Cartan's first structural equation,
$$d\omega^j=\omega^i \wedge \Theta^j_{\,\,i}.$$
By restricting to $\Sigma$ (and by the uniqueness of $\tilde{\Theta}$) we conclude that
$$\tilde{\Theta}^i_{\,\,j}=\Theta^i_{\,\,j} |_{T\Sigma}, \quad i,j=1,2,$$
and that
$$0= \tilde{\omega}^1\wedge\Theta^3_{\,\,1}|_{T\Sigma}+\tilde{\omega}^2\wedge\Theta^3_{\,\,2}|_{T\Sigma}.$$
So there exists (by Cartan lemma) smooth functions $s_{ij}$ defined on $\Sigma$, $i,j=1,2$, such that
$$\Theta_1^3|_{T\Sigma} = s_{11} \tilde{\omega}^1 + s_{12} \tilde{\omega}^2,$$
$$\Theta_2^3|_{T\Sigma} = s_{21} \tilde{\omega}^1 + s_{22} \tilde{\omega}^2,$$
with $s_{12}=s_{21}$.
The (0,2)-tensor $\text{II}=s_{ij} \tilde{\omega}^i\otimes \tilde{\omega}^j$ is called the second fundamental form, and its definition can be proven to be independent of the chosen frame. Observe that
$$\nabla_{e_1}e_3=e_1 \lrcorner \Theta^1_{\,\,3} e_1+e_1\lrcorner \Theta^2_{\,\,3} e_2+e_1 \Theta^3_{\,\,3} e_3=-s_{11}e_1-s_{21}e_2,$$
so $\text{II}(e_i,e_j)=-g(\nabla_{e_i} e_3,e_j)$.
It is the "lowering indices version" of the $(1,1)$-tensor called the shape operator or Weingarten map:
$$S(X)=-\nabla_X e_3$$
In term of the given frame the shape operator can be written as
$$S=\sum s_{ij}\tilde{\omega}^i\otimes \tilde{e}_j,$$

where $\tilde{e}_j=e_j|_{T\Sigma}$, $j=1,2$.
This new tensor has the following invarians:
- The semi-trace $H=\frac{1}{2}(s_{11}+s_{22})$, which is called the mean curvature.
- The determinant $K_{ext}=s_{11}s_{22}-s_{12}s_{21}$, which is the extrinsic Gauss curvature.

On the other hand, by Cartan's second structural equation, the curvature 2-forms $\Omega^i_{\,\, j}$ of $M$, defined by
$$\Omega^i_{\, j} = d \Theta^i_{\, j} + \Theta^i_{\, k} \wedge \Theta^k_{\, j}.$$
are related to the Riemann curvature tensor of $M$ by the equation
$$R^i_{\,jab}=\Omega^i_{\, j} (e_a,e_b) \tag{*}$$
in the given frame.  
Denote by $\tilde{R}$ the Riemann curvature tensor of $\Sigma$ with respect to the inherited metric. The curvature 2-form $\tilde{\Omega}$ of $\Sigma$ satisfies, by its definition,
$$\begin{aligned}     \tilde{\Omega}^i_{\,j}&=\Omega^i_{\,j}|_{T\Sigma}+\Theta_1^3|_{T\Sigma}\wedge \Theta_2^3|_{T\Sigma}\\     &=\Omega^i_{\,j}+K_{ext}\tilde{\omega}^1\wedge \tilde{\omega}^2. \end{aligned}$$
From here, and according to equation $(*)$, we obtain the well-known Gauss' equation,
$$K_{int}=R^1_{\,\,212}+K_{ext},$$
where $K_{int}:=\tilde{R}^1_{\,\,212}$ is the intrinsic Gaussian curvature of $\Sigma$.