Atlas / Mathematics / The Geometry Thread

Field · Emerged 18th century – 1820s

Differential Geometry of Surfaces

How do you measure the bending of a curve or surface with calculus, and how much of that bending can be detected from inside the surface?

5 chapters4 min read5 turning points 1 open problem

Branched from: Euclidean Geometry + Calculus
Branched into: Geometric Analysis + Riemannian Geometry
Figures: Leonhard Euler, Carl Friedrich Gauss, Jean Frédéric Frenet, Joseph Serret, Pierre Ossian Bonnet, David Hilbert

In brief

Differential geometry uses calculus to measure how curves and surfaces bend. A curve's bending is captured by two numbers at each point, and a surface's by two more. Gauss's central discovery was that the product of a surface's two curvatures can be measured by a creature living on the surface who never leaves it.

That split between how a shape bends in space and how it is curved in itself is the idea everything later in this thread builds on. It is also why no flat map of the Earth can be perfect.

Key ideas

Curvature and torsion of a curveEnters 1847 – 1851: Curvature $\kappa$ measures how fast a curve turns ( $1/r$ for the best-fitting circle). Torsion $\tau$ measures how fast it twists out of its plane. Together they determine a space curve completely.
Principal curvaturesEnters 1760 (published 1767): At each point of a surface, the largest and smallest bending among all directions, $\kappa_1$ and $\kappa_2$ . They always occur in perpendicular directions.
Gaussian curvatureEnters 1827: $K = \kappa_1\kappa_2$ . Positive on a sphere, zero on a cylinder or a flat sheet, negative on a saddle. Its sign alone tells you whether small triangles have angles summing to more, exactly, or less than $\pi$ .
Intrinsic vs. extrinsicEnters 1827: An intrinsic property can be measured from inside the surface, using only lengths and angles on it. An extrinsic one depends on how the surface sits in space. Gaussian curvature is intrinsic, and each principal curvature alone is not.
GeodesicEnters 1848: The straightest possible path on a surface, which locally is also the shortest. Great circles on a sphere are geodesics. They play the role of straight lines in the surface's own geometry.

Draws on other domains

↙ Biology
Membrane Biophysics
Red blood cells as a variational problem

Chapter I

Calculus Meets Shape

When Newton and Leibniz made calculus in the late seventeenth century, curvature was one of the first things it could measure. The curvature of a plane curve at a point is the reciprocal of the radius of the circle that best hugs it there: $\kappa = 1/r$ . Huygens had already used such circles to study pendulum clocks, and Clairaut soon extended the idea to curves twisting through space.

Surfaces were harder, because a surface curves differently in different directions. A cylinder is curved around its circumference and flat along its length. A saddle curves up one way and down the other. Leonhard Euler showed in 1760 that all of this is controlled by two numbers, the principal curvatures $\kappa_1$ and $\kappa_2$ , which occur in perpendicular directions. Gaspard Monge's school in France then built a whole descriptive theory of surfaces on that idea. For curves the story was essentially finished by the 1850s. Frenet and Serret showed that two functions, curvature $\kappa$ and torsion $\tau$ , determine any curve in space.

Up to here, this is still Euclid's space. Curves and surfaces are objects sitting in flat three-dimensional space, and their curvature describes how they bend within it.

Chapter II

The Remarkable Theorem

The turn came from surveying. In the 1820s Gauss ran a geodetic survey of the Kingdom of Hanover, measuring huge triangles on the curved Earth. That work led to his Disquisitiones generales circa superficies curvas (1827). In it, he described a surface entirely by how distances are measured on it, the first fundamental form:

ds^2 = E\,du^2 + 2F\,du\,dv + G\,dv^2 .

He then proved what he called the Theorema Egregium, the "remarkable theorem." The product $K = \kappa_1\kappa_2$ , now called the Gaussian curvature, can be computed from $E$ , $F$ , $G$ and their derivatives alone. Each principal curvature depends on how the surface sits in space, but their product does not. Rolling a sheet of paper into a cylinder changes $\kappa_1$ from $0$ to something nonzero, yet $K$ stays $0$ , which is why paper rolls without stretching. A sphere has $K > 0$ everywhere, which is why no flat map of the Earth can keep all distances correct. Gauss the surveyor had proved why every map must distort.

Chapter III

Intrinsic Geometry

The consequence was philosophical as much as technical: a surface has a geometry of its own, independent of any surrounding space. Pierre Ossian Bonnet made the link to angles explicit in 1848. For a triangle made of shortest paths (geodesics) on a surface,

\alpha + \beta + \gamma - \pi = \iint_T K\,dA .

Look at what this says. On a surface of constant negative curvature, the angles of every triangle add up to less than $\pi$ , and the shortfall is proportional to area. That is exactly the rule of the non-Euclidean geometry that Lobachevsky and Bolyai had just found by pure logic. The two new branches were describing the same thing from opposite sides.

Riemann was the one who joined them. His 1854 lecture took Gauss's intrinsic view, extended it to any number of dimensions, and made the non-Euclidean geometries special cases. That is where Riemannian geometry begins.

Chapter IV

A Closer Look: Measuring Curvature Without Leaving the Surface

The Theorema Egregium says curvature can be measured from inside a surface. Here is how a flat creature living on a sphere of radius $R$ could do it with nothing but a rope.

Fix one end of a rope of length $r$ at a point and walk around with the other end taut, tracing a circle. On a flat plane the circle's circumference would be $2\pi r$ . On the sphere, the "circle" is a line of latitude around the pole where the rope is fixed, and its circumference is

C(r) = 2\pi R \sin\!\left(\frac{r}{R}\right) = 2\pi r - \frac{\pi r^3}{3R^2} + \cdots ,

a little shorter than $2\pi r$ . The shortfall reveals the curvature. In general, Gauss's curvature is

K = \lim_{r \to 0} \frac{3}{\pi}\, \frac{2\pi r - C(r)}{r^3} ,

and for the sphere this gives $K = 1/R^2$ . On a saddle the circle comes out longer than $2\pi r$ , and $K$ is negative. On a cylinder it is exactly $2\pi r$ , and $K = 0$ , which is why a cylinder can be unrolled flat while a sphere cannot.

This is why every flat map of the Earth distorts. A map that kept all distances would carry the creature's rope-and-circle experiment over unchanged, but on paper the circle comes out $2\pi r$ and on the globe it comes out shorter. No choice of projection can reconcile the two.

Chapter V

Surfaces Today

Surfaces in ordinary space remain an active subject. The questions have moved on from "how curved is it?" to "what is the best possible shape?" Examples are minimal surfaces, the shapes of soap films, which locally have the least possible area, and surfaces that bend as little as possible overall. These problems sit where geometry meets nonlinear partial differential equations. Some of them were settled only in the last decade and a half.

Turning points5

1760 (published 1767)PROOF
Euler finds the two principal curvatures of a surface
Leonhard Euler studies how a surface curves as you turn in different directions from a point. He shows that every direction's curvature is set by just two extreme values, $\kappa_1$ and $\kappa_2$ : $\kappa(\theta) = \kappa_1 \cos^2\theta + \kappa_2 \sin^2\theta$ . Surface curvature becomes something calculus can measure.
Leonhard Euler
› Sources (1)
- Euler, L. (1767). Recherches sur la courbure des surfaces. Mémoires de l'académie des sciences de Berlin 16: 119–143.
1827PROOF
The Theorema Egregium — curvature is intrinsic
Gauss proves that the product $K = \kappa_1 \kappa_2$ can be computed from measurements made entirely within the surface: lengths and angles, with no reference to the space around it. A creature living on the surface could measure its curvature without ever leaving it.
Carl Friedrich Gauss
› Sources (1)
- Gauss, C. F. (1828). Disquisitiones generales circa superficies curvas. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 6. (Presented 1827.)
1847 – 1851PROOF
The Frenet–Serret formulas describe every space curve
Jean Frédéric Frenet, in his 1847 thesis, and Joseph Serret, in 1851, independently derive the formulas that govern a moving frame along a curve. Two functions, curvature $\kappa$ and torsion $\tau$ , determine the curve completely up to a rigid motion. The theory of curves in space was essentially finished, and surfaces became the frontier.
Jean Frédéric Frenet, Joseph Serret
› Sources (2)
- Frenet, F. (1852). Sur les courbes à double courbure. Journal de Mathématiques Pures et Appliquées 17: 437–447.
- Serret, J. A. (1851). Sur quelques formules relatives à la théorie des courbes à double courbure. Journal de Mathématiques Pures et Appliquées 16: 193–207.
1848PROOF
The Gauss–Bonnet theorem ties curvature to angles
Pierre Ossian Bonnet generalises a result Gauss had proved for triangles made of shortest paths. The total curvature inside a region controls how far its angles depart from the Euclidean values. Curvature and angle sums are two faces of the same quantity, and non-Euclidean geometry is revealed as the geometry of constant curvature.
Pierre Ossian Bonnet
› Sources (1)
- Bonnet, O. (1848). Mémoire sur la théorie générale des surfaces. Journal de l'École Polytechnique 19: 1–146.
1901PROOF
The full hyperbolic plane does not fit in ordinary space
Hilbert proves that no complete, smooth surface of constant negative curvature can sit inside three-dimensional Euclidean space. Beltrami's pseudosphere can only ever show a piece of the hyperbolic plane. The whole thing has to be studied from the inside, which is one more push toward Riemann's intrinsic point of view.
David Hilbert
› Sources (1)
- Hilbert, D. (1901). Über Flächen von konstanter Gaussscher Krümmung. Transactions of the AMS 2(1): 87–99.

Applications

Where it is used

Cartography
Why every map projection distorts
The Theorema Egregium says a sphere ( $K > 0$ ) cannot be flattened onto a plane ( $K = 0$ ) without changing some distances. Every map projection is therefore a choice of what to sacrifice. Mercator keeps angles and inflates areas near the poles. Equal-area projections keep areas and distort shapes. Geodesy, the science of measuring the Earth, still runs on Gauss's surface theory.
› Sources (1)
- Snyder, J. P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395.
Computer graphics
Curvature on digital meshes
3D models are meshes of tiny triangles, and graphics software estimates their curvature to smooth noise, simplify models and place detail where the surface bends most. Car and product designers demand surfaces whose curvature varies smoothly across seams, because reflections reveal every kink.
› Sources (1)
- Botsch, M., Kobbelt, L., Pauly, M., Alliez, P. & Lévy, B. (2010). Polygon Mesh Processing. A K Peters.
Architecture
Building curved forms from flat sheets
A surface with zero Gaussian curvature, a developable surface, can be bent from a flat sheet without stretching. Architects and engineers use this to make curved facades from sheet metal or glass. Doubly curved panels ( $K \neq 0$ ) need costly moulds.
› Sources (1)
- Pottmann, H., Asperl, A., Hofer, M. & Kilian, A. (2007). Architectural Geometry. Bentley Institute Press.
Cell biology↗ Biology · Membrane Biophysics
The shape of membranes and red blood cells
A cell membrane resists bending, and its elastic energy is an integral of squared mean curvature, the same quantity as the Willmore energy below. Minimising it explains the biconcave disc shape of human red blood cells, as Canham (1970) and Helfrich (1973) showed.
› Sources (2)
- Canham, P. B. (1970). The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology 26(1): 61–81.
- Helfrich, W. (1973). Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift für Naturforschung C 28: 693–703.
Optics↗ Physics · Nonlinear and Nano-Optics
Transformation optics
Maxwell's equations keep their form under a change of coordinates if the material properties are transformed accordingly, so a desired bending of light can be specified as a coordinate map and then realised as a prescription for a material's permittivity and permeability. Designs for cloaks and flat lenses are produced this way: the geometry is chosen first and the metamaterial is derived from it.
› Sources (2)
- Pendry, J. B., Schurig, D. & Smith, D. R. (2006). Controlling electromagnetic fields. Science 312: 1780–1782.
- Leonhardt, U. (2006). Optical conformal mapping. Science 312: 1777–1780.

Open problems

Where the map runs out

Open

Local isometric embedding of surfaces

Open in general when the curvature changes sign in a degenerate way.

Take any smooth surface described only by its internal distances. Can a small patch of it always be built as an actual surface in ordinary three-dimensional space with exactly those distances? The answer is yes when the Gaussian curvature is positive, negative, or crosses zero cleanly. When the curvature vanishes in a degenerate way, no one knows.

Why it is hard: The problem becomes a nonlinear partial differential equation whose type depends on the sign of the curvature. Where the curvature is positive it behaves like one kind of equation, where negative like another, and on the boundary between them neither body of standard technique applies. Counterexamples exist for metrics that are only finitely smooth, so any proof must use full smoothness in an essential way.
What resolving it unlocks: It would settle whether Gauss's intrinsic view and the classical picture of surfaces sitting in space always agree at small scales. The methods would carry over to other PDEs that change type.

› Sources (1)

Han, Q. & Hong, J.-X. (2006). Isometric Embedding of Riemannian Manifolds in Euclidean Spaces. American Mathematical Society.

Recently resolved

The Willmore conjecture

Proved by Fernando Codá Marques and André Neves in 2012, published 2014.

Thomas Willmore asked in 1965 how little a torus (a doughnut-shaped surface) can bend. He measured bending by the total squared mean curvature and conjectured that every torus in space has $\int H^2\,dA \ge 2\pi^2$ , with the minimum reached by one specific torus of revolution.

Why it is hard: The Willmore energy does not change under a large family of transformations, the conformal maps, including inversions in spheres. A sequence of tori whose energy keeps falling can therefore degenerate in ways that are hard to control. Marques and Neves had to revive Almgren–Pitts min-max theory, a decades-old method for finding minimal surfaces, to rule every such escape out.
What resolving it unlocks: Its main legacy is the min-max method it brought back. That method soon settled Yau's conjecture that every closed three-manifold contains infinitely many minimal surfaces (Irie, Marques and Neves for generic metrics, then Antoine Song in full).

› Sources (2)

Willmore, T. J. (1965). Note on embedded surfaces. Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi 11B: 493–496.
Marques, F. C. & Neves, A. (2014). Min-max theory and the Willmore conjecture. Annals of Mathematics 179(2): 683–782.

In brief

Key ideas

Draws on other domains

Calculus Meets Shape

The Remarkable Theorem

Intrinsic Geometry

A Closer Look: Measuring Curvature Without Leaving the Surface

Surfaces Today

Where it is used

Why every map projection distorts

Curvature on digital meshes

Building curved forms from flat sheets

The shape of membranes and red blood cells

Transformation optics

Where the map runs out

Local isometric embedding of surfaces

The Willmore conjecture

Further reading