Gauss' equation

General setup

See @spivak1999comprehensive volume 3, page 5.
It is a generalization of Theorema Egregium. Given an immersed Riemannian submanifold $M$ in an ambient Riemannian manifold $N$ , it provides a relation between the intrinsic curvature of $M$ , the extrinsic curvature of $M$ and the curvature of $N$ . Given the orthogonal complement $T_{p} M^{⊥} \subset T_{p} N$ , we can use the decomposition $T_{p} N = T_{p} M \oplus T_{p} M^{⊥}$ to define two projections:

\begin{aligned} T : & T_{p} N \to T_{p} M & (the tangential projection) \\ ⊥: & T_{p} N \to T_{p} M^{⊥} & (the normal, or perpendicular, projection) \end{aligned}

with

X = T X + ⊥ X for all X \in T_{p} N .

First, observe the following theorem (@spivak1999comprehensive volume 3, page 2):
Theorem 1
Let $i : M \to N$ be an immersion, where $N$ is a Riemannian manifold and $M$ has the induced metric, and denote by $\nabla$ and $\nabla^{'}$ the covariant derivatives for $M$ and $N$ , respectively. If $X_{p} \in T_{p} M$ , and $Y$ is a vector field on $M$ which is everywhere tangent to $M$ , then

\nabla_{X_{p}} Y = T (\nabla_{X_{p}}^{'} Y),

where $T : T_{p} N \to T_{p} M$ is the usual projection. $◼$

On the other hand, notice that

⊥ (\nabla_{X}^{'} f Y) =⊥ (X_{p} (f) \cdot Y_{p} + f (p) \cdot \nabla_{X}^{'} Y) = f (p) \cdot ⊥ (\nabla_{X}^{'} Y) .

So we can define the tensor field $s : M_{⊥} \times M_{⊥} \to M_{⊥}$ for each $p \in M$ , such that for any vector field $Y$ extending $Y_{p}$ :

s (X_{p}, Y_{p}) =⊥ (\nabla_{X}^{'} Y),

where $T N = T M \oplus M_{⊥}$ and for $V \in T_{p} N$ we have $V = T (V) \oplus ⊥ (V)$ , for $p \in M$ .

Theorem. The tensor $s$ is symmetric (Theorem 5 in @spivak1999comprehensive):
Proof
Let $X$ and $Y$ be any extensions of $X_{p}, Y_{p} \in T_{p} M$ to all of $N$ which are tangent to $M$ at all points of $M$ . Then

⊥ (\nabla_{X}^{'} Y) - ⊥ (\nabla_{Y}^{'} X) =⊥ (\nabla_{X}^{'} Y_{p} - \nabla_{Y}^{'} X) =⊥ (\nabla_{X}^{'} Y (p) - \nabla_{Y}^{'} X (p)) =⊥ ([X, Y] (p)) = 0,

since $[X, Y]$ is also tangent to $M$ at all points of $M$ . $◼$

The tensor $s$ is a kind of second fundamental form, see below.

So we have Gauss' formula:

\nabla_{X_{p}}^{'} Y = \nabla_{X_{p}} Y + s (X_{p}, Y_{p}),

and Gauss' equation (generalization of Theorema Egregium):
Theorem 6. Let $R^{'}$ and $R$ denote the Riemann curvature tensors of $M$ and $N$ , respectively. For all $X_{p}, Y_{p}, Z_{p}, W_{p} \in T_{p} M$ we have:

⟨ R^{'} (X_{p}, Y_{p}) Z_{p}, W_{p} ⟩ = ⟨ R (X_{p}, Y_{p}) Z_{p}, W_{p} ⟩ + ⟨ s (X_{p}, Z_{p}), s (Y_{p}, W_{p}) ⟩ - ⟨ s (Y_{p}, Z_{p}), s (X_{p}, W_{p}) ⟩

$◼$

For hypersurfaces

Let $M$ be a hypersurface in $N$ , and let $ν$ be a unit normal field on a neighborhood of $p$ in $M$ . We define

I I (X_{p}, Y_{p}) := ⟨ ν, s (X_{p}, Y_{p}) ⟩ .

It turns out that in case of a surface in $R^{3}$ , this is precisely the second fundamental form.
Theorem 8. (@spivak1999comprehensive page 7)
Let $M$ be a hypersurface in $N$ , and let $ν$ be a unit normal field on a neighborhood of $p$ in $M$ .
(a) For all $X_{p} \in T_{p} M$ (where $X_{p}$ is a vector in the tangent space of $M$ at $p$ ) we have:

\nabla_{X_{p}}^{'} ν \in T_{p} M .

(b) If $Y$ is a vector field tangent along $M$ , then we have the Weingarten Equations:

⟨ \nabla_{X_{p}}^{'} ν, Y_{p} ⟩ = - ⟨ ν, \nabla_{X_{p}}^{'} Y ⟩ = - ⟨ ν, s (X_{p}, Y_{p}) ⟩ .

⟨ \nabla_{X_{p}}^{'} ν, Y_{p} ⟩ = ⟨ X_{p}, \nabla_{Y_{p}}^{'} ν ⟩ .

$◼$
According to this theorem, this second fundamental form $I I (X_{p}, -)$ is a kind of shape operator, $\nabla_{X_{p}}^{'} ν$ , since for an orthonormal frame ${e_{i}}$ in $M$ we have:

I I (X_{p}, e_{i}) = ⟨ ν, s (X_{p}, e_{i}) ⟩ = - ⟨ \nabla_{X_{p}}^{'} ν, e_{i} ⟩ .

Closely related, we have the Weingarten map:

X_{p} \mapsto \sum - I I (X_{p}, e_{i}) e_{i} = \sum ⟨ \nabla_{X_{p}}^{'} ν, e_{i} ⟩ e_{i}

I think that from here we could define principal curvatures and the extrinsic curvature of the hypersurface...

In case $d i m (M) = 2$ , we can apply Gauss' equation above to ${e_{1}, e_{2}}$ to obtain:

⟨ R^{'} (e_{1}, e_{2}) e_{2}, e_{1} ⟩ = ⟨ R (e_{1}, e_{2}) e_{2}, e_{1} ⟩ + ⟨ s (e_{1}, e_{2}), s (e_{2}, e_{1}) ⟩ - ⟨ s (e_{2}, e_{2}), s (e_{1}, e_{1}) ⟩

Recall that $⟨ R (e_{1}, e_{2}) e_{2}, e_{1} ⟩$ is the sectional curvature, so this equation can be rewritten as

K_{s e c t} = K - K_{e x t},

where:

$K_{s e c t}$ is the Gauss curvature of a surface $M^{'}$ made of geodesics and tangent to ${e_{1}, e_{2}}$ ,
$K$ is the Gauss curvature, in the sense of holonomy, of the surface $M$ .
$K_{e x t}$ is the extrinsic curvature (recall that it is the determinant of the Weingarten map).

For surfaces in $R^{3}$

In the case of $R^{3}$ , the sectional curvature is always 0, since the geodesic surfaces are planes. So we have

K = K_{e x t} .

Via moving frames

See geometry of a submanifold via moving frames.

Relation with the curvature of the connection

The Gauss equation plays a crucial role in differential geometry, particularly in the study of surfaces. It relates the intrinsic curvature of a surface to its extrinsic curvature, encapsulating how the surface bends in the ambient space. For an immersed surface $ι : S \to R^{3}$ , we consider a local orthonormal frame ${e_{1}, e_{2}}$ in a neighborhood of a point $p \in S$ , ${ω^{1}, ω^{2}}$ is the dual coframe and $Θ$ is the connection 1-form matrix. Gauss equation says that

d Θ_{2}^{1} = K ω^{1} \land ω^{2}

(@ivey2016cartan page 47) where $K$ is the Gaussian curvature of the surface.

-Proof of Gauss equation.-
(Also in @needham2021visual section 38.5.1). Consider a point $p \in S$ and an open neighborhood $U \subset R^{3}$ of $p$ . We have the unit normal vector to the surface $n$ , and think of a orthonormal frame ${{\tilde{e}}_{1}, {\tilde{e}}_{2}, \tilde{n}}$ defined on $U$ such that gives rise to ${e_{1}, e_{2}, n}$ when restricted to $S$ . Analogously, we have the dual coframe ${{\tilde{ω}}_{1}, {\tilde{ω}}_{2}, \tilde{ζ}}$ . We can compute here the connection 1-forms matrix $\tilde{Θ}$ of the standard connection $\nabla_{s t d}$ with respect to this frame (see this). It turns out that

\tilde{Θ} = (\begin{array}{cccc} 0 & {\tilde{Θ}}_{2}^{1} & {\tilde{Θ}}_{3}^{1} \\ - {\tilde{Θ}}_{2}^{1} & 0 & {\bar{Θ}}_{3}^{2} \\ - {\bar{Θ}}_{3}^{1} & - {\bar{Θ}}_{3}^{2} & 0 \end{array})

where it is satisfied that $ι^{*} ({\tilde{Θ}}_{2}^{1}) = Θ_{2}^{1}$ . To see this, observe that on the one hand we have

\nabla_{V} e_{2} = V ⌟ Θ_{2}^{1} e_{1},

and in the other, since $\nabla$ is the induced connection by the standard metric in $R^{3}$ ,

\nabla_{V} e_{2} = proj ((\nabla_{s t d})_{V} {\tilde{e}}_{2}) = proj (V ⌟ {\tilde{Θ}}_{2}^{1} {\tilde{e}}_{1} - V ⌟ {\tilde{Θ}}_{3}^{2} \tilde{n}) = V ⌟ ι^{*} ({\tilde{Θ}}_{2}^{1}) e_{1}

for an arbitrary vector $V$ tangent to $S$ .

Now, we have to put together several things:

First, we know that Gaussian curvature $K$ is the determinant of the shape operator.
Second, it turns out that this determinant can be computed as ${\tilde{Θ}}_{3}^{1} \land {\tilde{Θ}}_{3}^{2} (e_{1}, e_{2})$ (see here to see why) evaluated in points $p$ of the surface $S$ .
Then, according to Cartan's second structural equation, since $R^{3}$ is flat

d {\tilde{Θ}}_{j}^{i} = - {\tilde{Θ}}_{k}^{i} \land {\tilde{Θ}}_{j}^{k}

and then

d {\tilde{Θ}}_{2}^{1} = {\tilde{Θ}}_{2}^{3} \land {\tilde{Θ}}_{3}^{1}

and

d {\tilde{Θ}}_{2}^{1} (e_{1}, e_{2}) = {\tilde{Θ}}_{2}^{3} \land {\tilde{Θ}}_{3}^{1} (e_{1}, e_{2}) = - {\tilde{Θ}}_{3}^{1} \land {\tilde{Θ}}_{2}^{3} (e_{1}, e_{2}) = {\tilde{Θ}}_{3}^{1} \land {\tilde{Θ}}_{3}^{2} (e_{1}, e_{2})

If we restrict this identity to $S$ we obtain

d Θ_{2}^{1} (e_{1}, e_{2}) = K .

Pending task: I want to describe all latter stuff with the method of moving frames. I think I can show that $K$ is one of the invariants provided by the MC form, but for the elements of Euclidean group $E (3)$ . So $K$ is invariant under congruence (like median curvature, I guess). But I don't know yet how to show it for general isometries of the surface. It is maybe explained here. May is also interesting this paper of Chern. (See also @ivey2016cartan page 47)
Maybe this is also related to what is said in Generalization of the flatness of R3.