= Theorema Egregium
{c}
{title2=$K_N\circ f=K_M$}
{wiki}
For <smooth surfaces>, <Gaussian curvature> depends only on the <first fundamental form> and its first two coordinate <derivatives>. Consequently any smooth <local isometry> $f:M\to N$ satisfies $K_N(f(p))=K_M(p)$. The surprise is that $K$ is defined using the <shape operator>, which describes bending in space, yet its value can be recovered from measurements made entirely within the surface.
In his 1827 paper, <Carl Friedrich Gauss> expressed <Gaussian curvature> through the metric coefficients and their <partial derivatives> in Article 11, then stated its invariance under development in Article 12. Article 13 explicitly compares a <plane> with surfaces developable onto it, including cylindrical surfaces. The paper was presented on 8 October 1827 and printed in 1828; the Morehead–Hiltebeitel English translation appeared in 1902. https://www.gutenberg.org/files/36856/36856-pdf.pdf[Gauss's original paper in English translation, Articles 11–13 and title pages]{ref}
A <plane> and a <circular cylinder> can have different <principal curvatures> while both have $K=0$. Their matching <first fundamental forms> exhibit the actual <local isometry>. In contrast, no nonempty open patch of a <sphere> of radius $R$ can be flattened smoothly into a <plane> while preserving its <first fundamental form>: $1/R^2$ cannot become zero. Making the patch small reduces practical distortion, but never produces an exact <local isometry>. This is a geometric obstruction to a perfectly distance-preserving flat map of a spherical world, independent of where one places a seam.
The following proof outline connects the two descriptions of <Gaussian curvature>. For a <parametric 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 vector> <inner products> determines the lowered <Christoffel symbols> solely from the <Riemannian 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
$$
K=\frac{h_{11}h_{22}-h_{12}^2}{\det(g_{ij})}
=\frac{R_{1212}}{\det(g_{ij})},
$$
where $R_{1212}$ is the corresponding component of the intrinsic <Riemann curvature tensor>, with the sign convention in the displayed <Gauss equation>. The final expression uses only $g_{ij}$ and its first two <derivatives>.
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