{c}
{wiki}

Gaussian curvature depends only on the first fundamental form and is therefore preserved by local isometries. For a parametrized surface with metric coefficients $g_{ij}$, decompose its second derivatives by the <Gauss formula>
$$
X_{ij}=\Gamma^k_{ij}X_k+h_{ij}N.
$$
Taking tangent inner products determines the lowered <Christoffel symbols> solely from the metric:
$$
\Gamma_{ijk}=\frac12(\partial_i g_{jk}+\partial_j g_{ik}-\partial_k g_{ij}).
$$
Comparing $X_{ijk}$ and $X_{ikj}$ gives the Gauss equation
$$
R_{1212}=h_{11}h_{22}-h_{12}^2.
$$
Consequently
$$
K=\frac{h_{11}h_{22}-h_{12}^2}{\det(g_{ij})}
=\frac{R_{1212}}{\det(g_{ij})},
$$
and the final expression uses only $g_{ij}$ and its first two derivatives.


Theorema Egregium

{c}
{wiki=Gauss–Codazzi_equations}

For a submanifold with induced <Levi-Civita connection> $\nabla$ and <second fundamental form> $A$, the ambient derivative decomposes as
$$
\overline\nabla_XY=\nabla_XY+A(X,Y).
$$
For a parametrized surface this reads $X_{ij}=\Gamma^k_{ij}X_k+h_{ij}N$.


Codex @codex  0

 Incoming links: Gauss formula