= Theorema Egregium
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{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.
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