Intrinsic Curvature: Surface Geometry and the Theorema Egregium

Intrinsic vs Extrinsic: Two Views of Curvature

A surface may be understood in two distinct ways. Extrinsic properties depend on how the surface sits in ambient three-dimensional space—they require an “outside” view. The normal vector perpendicular to the surface points into 3D space, invisible to a two-dimensional being confined to the surface. Mean curvature, the average of principal curvatures, changes when you bend a surface, even if distances along the surface remain unchanged.

Intrinsic properties, by contrast, are measurable by an inhabitant who never leaves the surface. Geodesics—the shortest paths between points—are intrinsic. Angles measured along the surface are intrinsic. Most importantly, Gaussian curvature $K$ is intrinsic, as my Theorema Egregium proved.

Consider a sphere and a cylinder. Extrinsically, they differ profoundly: the sphere is closed and convex; the cylinder is open with straight lines running along its length. Yet intrinsically, they share a fundamental property. The cylinder has Gaussian curvature $K = 0$ everywhere, just like a plane. Why? Because you can roll a cylinder flat onto a plane without stretching or tearing the material—an isometry exists. A two-dimensional being living on the cylinder would measure the same distances and angles as one living on a plane.

An isometry is a distance-preserving map. If the distance $d(P,Q)$ on surface 1 equals $d(f(P), f(Q))$ on surface 2 for all points $P$ and $Q$, then $f$ is an isometry and preserves intrinsic geometry. My profound insight in the 1820s was recognizing that intrinsic geometry is independent of embedding. A 2D being could discover the curvature of their world without accessing a third dimension. By measuring geodesic triangles, they would find that the sum of angles equals $\pi$ for flat space, exceeds $\pi$ for positive curvature, and falls short of $\pi$ for negative curvature.

Theorema Egregium: Curvature Without Embedding

At each point on a surface, curves passing through that point in different directions have different curvatures. The normal curvature $\kappa_n(\theta)$ varies with direction $\theta$. The maximum and minimum values are the principal curvatures $\kappa_1$ and $\kappa_2$, occurring in perpendicular directions called the principal directions.

Gaussian curvature is defined as $K = \kappa_1 \kappa_2$, the product of principal curvatures. This product carries deep information about the surface’s geometry.

For a sphere of radius $r$, all directions curve equally: $\kappa_1 = \kappa_2 = 1/r$. The curvature is isotropic, and $K = 1/r^2$, positive and constant everywhere.

For a cylinder of radius $r$, one direction curves around the circumference with $\kappa_1 = 1/r$, while the other direction runs straight along the generator with $\kappa_2 = 0$. Thus $K = 0$. The cylinder is intrinsically flat.

For a saddle or hyperboloid, the surface curves upward in one direction ($\kappa_1 > 0$) and downward in the perpendicular direction ($\kappa_2 < 0$). The product $K < 0$ gives negative curvature, characteristic of hyperbolic geometry.

For a plane, both principal curvatures are zero: $\kappa_1 = \kappa_2 = 0$, so $K = 0$—Euclidean geometry.

Mean curvature $H = (\kappa_1 + \kappa_2)/2$ is extrinsic. It changes under bending and is minimized by soap films, which are minimal surfaces with $H = 0$. For a sphere, $H = 1/r$.

My Theorema Egregium—the “Remarkable Theorem”—proved in 1827 that Gaussian curvature $K$ is intrinsic. It can be computed from the metric tensor $g_{ij}$, which encodes distances and angles on the surface, without any reference to how the surface sits in three-dimensional space. The formula is $K = R_{1212}/\det(g)$, where $R$ is the Riemann curvature tensor defined purely from the metric, without normal vectors.

The consequence is profound: bending preserves $K$. An isometry maintains distances, and since $K$ is intrinsic, it remains unchanged. This is why you cannot smooth an orange peel flat without tearing it. The sphere has $K > 0$ everywhere, while the plane has $K = 0$. No isometry can exist between them.

Why You Can’t Flatten an Orange Peel

The cartographer’s problem is representing Earth—approximately a sphere with $K = 1/r^2 \approx 2.5 \times 10^{-14} \text{ m}^{-2}$—on a flat map where $K = 0$. By my theorem, this is impossible while preserving both angles and areas. An isometry would preserve $K$, but the sphere and plane have different curvatures. No such map exists.

This forces cartographers to choose. Conformal projections preserve angles locally. The Mercator projection, cylindrical in construction, maintains angles but grotesquely distorts areas near the poles—Greenland appears larger than Africa though it is actually one-fourteenth the size. The stereographic projection, projecting from a pole onto a plane, also preserves angles and maps circles to circles.

Equal-area projections preserve areas but distort shapes. The Mollweide projection, elliptical in appearance, maintains area ratios but warps shapes near the edges. The Lambert azimuthal projection preserves areas radially from a chosen center but distorts angles.

Compromise projections minimize overall distortion without being strictly conformal or equal-area. The Robinson projection, used by National Geographic from 1988 to 1998, and the Winkel tripel, their current standard, attempt to balance competing demands.

The Gauss-Bonnet theorem connects curvature to topology: $\iint K \, dA = 2\pi \chi$, where $\chi$ is the Euler characteristic. For a sphere, $\chi = 2$. For a torus, $\chi = 0$. For a genus-$g$ surface, $\chi = 2 - 2g$. Total curvature equals topology, which cannot change under smooth deformations. This explains why the sphere’s non-zero total curvature prevents flattening.

Geodesic triangles on a sphere have angle sum $\alpha + \beta + \gamma = \pi + A/r^2$, where $A$ is the triangle’s area. The excess over $\pi$ equals the area times the curvature. Surveyors measuring large triangles detect Earth’s curvature this way. I performed such measurements during my Hanover geodetic survey in the 1820s, developing differential geometry tools to analyze the data.

From Surveying Earth to Curved Spacetime

Albert Einstein’s general relativity, formulated in 1915, extended my work to four dimensions and reinterpreted gravity as spacetime curvature—intrinsic geometry with no external embedding required.

The metric tensor $g_{\mu\nu}$ defines distances in spacetime: $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$, generalizing the Pythagorean theorem. Proper time along a worldline integrates this metric.

The Riemann curvature tensor $R_{\mu\nu\rho\sigma}$ measures curvature in four dimensions, with 20 independent components. In two dimensions, this reduces to the single Gaussian curvature.

Einstein’s field equations state $G_{\mu\nu} = 8\pi T_{\mu\nu}$, where the Einstein tensor $G_{\mu\nu}$ contains Riemann curvature, and the stress-energy tensor $T_{\mu\nu}$ represents matter and energy. Matter tells spacetime how to curve; spacetime tells matter how to move.

The Schwarzschild solution describes spherically symmetric black holes. Spatial slices exhibit Gaussian curvature scaling as $K \propto 1/r^4$ near the singularity, diverging to infinity at $r = 0$.

In cosmology, the Friedmann equations govern universal expansion. Critical density $\rho_c$ determines spatial curvature: if $\rho > \rho_c$, we have $K > 0$ and a closed universe; if $\rho < \rho_c$, we have $K < 0$ and an open universe; if $\rho = \rho_c$, we have $K = 0$ and a flat universe. Observations suggest $K \approx 0$—our universe appears spatially flat on large scales.

My Theorema Egregium anticipated general relativity. The concept of intrinsic curvature without embedding, developed for two-dimensional surfaces, became the foundation for Bernhard Riemann’s extension to $n$ dimensions. Einstein applied this to spacetime, treating it as a four-dimensional pseudo-Riemannian manifold with Lorentzian signature. The seeds I planted in 19th-century surveying grew into the relativistic description of the cosmos itself.