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 \(\Sigma\).

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