Geometry of submanifolds via moving frames

Consider, for simplicity, a 2-dimensional manifold $Σ$ 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 $Σ$ , i.e., $ω^{3} |_{T Σ} = 0$ , where $ω^{1}, ω^{2}, ω^{3}$ is the dual coframe.
The surface $Σ$ inherits a Riemannian metric from the ambient manifold, with its corresponding Levi-Civita connection. We will denote by ${\tilde{ω}}^{1}, {\tilde{ω}}^{2}$ the restrictions of $ω^{1}, ω^{2}$ to $T Σ$ , and by $\tilde{Θ}$ and $\tilde{Ω}$ the connection forms and the curvature forms, respectively, of the inherited connection.

According to Cartan's first structural equation,

d ω^{j} = ω^{i} \land Θ_{i}^{j} .

By restricting to $Σ$ (and by the uniqueness of $\tilde{Θ}$ ) we conclude that

{\tilde{Θ}}_{j}^{i} = Θ_{j}^{i} |_{T Σ}, i, j = 1, 2,

and that

0 = {\tilde{ω}}^{1} \land Θ_{1}^{3} |_{T Σ} + {\tilde{ω}}^{2} \land Θ_{2}^{3} |_{T Σ} .

So there exists (by Cartan lemma) smooth functions $s_{i j}$ defined on $Σ$ , $i, j = 1, 2$ , such that

Θ_{1}^{3} |_{T Σ} = s_{11} {\tilde{ω}}^{1} + s_{12} {\tilde{ω}}^{2},

Θ_{2}^{3} |_{T Σ} = s_{21} {\tilde{ω}}^{1} + s_{22} {\tilde{ω}}^{2},

with $s_{12} = s_{21}$ .
The $(0, 2)$ -tensor $II = s_{i j} {\tilde{ω}}^{i} \otimes {\tilde{ω}}^{j}$ 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} ⌟ Θ_{3}^{1} e_{1} + e_{1} ⌟ Θ_{3}^{2} e_{2} + e_{1} Θ_{3}^{3} e_{3} = - s_{11} e_{1} - s_{21} e_{2},

so $II (e_{i}, e_{j}) = - g (\nabla_{e_{i}} e_{3}, e_{j})$ .
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_{i j} {\tilde{ω}}^{i} \otimes {\tilde{e}}_{j},

where ${\tilde{e}}_{j} = e_{j} |_{T Σ}$ , $j = 1, 2$ .
This new tensor has the following invariants:

The semi-trace $H = \frac{1}{2} (s_{11} + s_{22})$ , which is called the mean curvature.
The determinant $K_{e x t} = s_{11} s_{22} - s_{12} s_{21}$ , which is the extrinsic curvature.

On the other hand, by Cartan's second structural equation, the curvature 2-forms $Ω_{j}^{i}$ of $M$ , defined by

Ω_{j}^{i} = d Θ_{j}^{i} + Θ_{k}^{i} \land Θ_{j}^{k} .

are related to the Riemann curvature tensor of $M$ by the equation

\begin{matrix} (*) & R_{j a b}^{i} = Ω_{j}^{i} (e_{a}, e_{b}) \end{matrix}

in the given frame.
Denote by $\tilde{R}$ the Riemann curvature tensor of $Σ$ with respect to the inherited metric. The curvature 2-form $\tilde{Ω}$ of $Σ$ satisfies, by its definition,

\begin{aligned} {\tilde{Ω}}_{j}^{i} & = Ω_{j}^{i} |_{T Σ} + Θ_{1}^{3} |_{T Σ} \land Θ_{2}^{3} |_{T Σ} \\ = Ω_{j}^{i} + K_{e x t} {\tilde{ω}}^{1} \land {\tilde{ω}}^{2} . \end{aligned}

From here, and according to equation $(*)$ , we obtain the well-known Gauss' equation,

K_{i n t} = R_{212}^{1} + K_{e x t},

where $K_{i n t} := {\tilde{R}}_{212}^{1}$ is the intrinsic Gaussian curvature of $Σ$ .