THE GRADUATE STUDENT SECTION

WHAT IS. . .

Gauss Curvature?Editors

Gaussian curvature is a curvature intrinsic to a two-dimensional surface, something you’d never expect asurface to have. A bug living inside a curve cannot tell ifit is curved or not; all the bug can do is walk forward andbackward, measuring distance. But a very intelligent bugliving on a surface can, by walking around, never leavingthe surface, measure its Gaussian curvature.

The Definition. Assuming you know how to define thecurvature of a curve, how would you define the curvatureof a surface at a point 𝑝? Slice the surface by a planenormal to the surface at 𝑝, as in Figure 1.

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Figure 1. To define the Gauss curvature, consider thecurvature of slice curves. The Gauss curvature is theproduct of the largest (positive) and the smallest ormost negative such curvatures.

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Take the curvature of the slice-curve. Thus you get acurvature in every direction. The largest (positive, upward)and smallest (most negative) such curvatures are calledthe principal curvatures, and they occur in orthogonaldirections. Their average is called the mean curvature,and their product is called the Gauss curvature. For theunit sphere, both principal curvatures are 1 and hencethe Gauss curvature is 1. For a unit cylinder, the principalcurvatures are 1 and 0 and hence the Gauss curvatureis 0. For a suitable hyperbolic paraboloid as in Figure 2,the principal curvatures are 1 and −1, and the Gausscurvature is −1.

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Figure 2. For this hyperbolic paraboloid, the Gausscurvature is (+1)(−1) = −1(+1)(−1) = −1(+1)(−1) = −1.

Gauss’s Theorema Egregium says that this Gauss cur-vature is intrinsic; in other words, it can be measured bya bug within the surface. That means that if you bend thesurface without stretching it, the Gauss curvature cannotchange. If you roll the plane up into a cylinder, as inFigure 3, the principal curvatures change from 0, 0 to 0, 1,but the Gauss curvature remains 0.

144 Notices of the AMS Volume 63, Number 2

THE GRADUATE STUDENT SECTIONIm

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Figure 3. Rolling a piece of plane into a cylinder ofradius 𝑟 changes the principal curvatures from 0, 00, 00, 0 to1/𝑟, 01/𝑟, 01/𝑟, 0 but the Gauss curvature–the product–remains 0.

One way a bug can measure the Gauss curvature ofa surface at a point 𝑝 is by looking at a tiny “geodesic”circle of tiny radius 𝑟 about 𝑝:

{𝑥 ∶ dist(𝑝, 𝑥) = 𝑟}.If the Gauss curvature is 0, the circumference is 2𝜋𝑟to high order. If the Gauss curvature is positive as onthe sphere, the circumference is smaller. If the Gausscurvature is negative as on the hyperbolic paraboloid, thecircumference is greater. In general, for Gauss curvature𝐺, the circumference is given by

2𝜋𝑟−𝐺𝜋𝑟3

3 +….

For a closed surface, there is the amazing Gauss-Bonnet formula relating the Gauss curvature 𝐺 andthe Euler characteristic 𝜒. The Euler characteristic is apurely combinatorial quantity, easily computed from anytriangulation as the number of vertices minus the numberof edges plus the number of regions. The Gauss-BonnetTheorem says that

∫𝐺 = 2𝜋𝜒.

Since the Euler characteristic of the sphere is 2, this saysthat the integral of the Gaussian curvature is 4𝜋. This iseasy to check for the round unit sphere: since the Gausscurvature is 1, it just says that the area is 4𝜋. The amazingthing is that it remains true for a deformed sphere of anysize and shape.

The Gauss-Bonnet Theorem is an amazing equalityof the geometric and the combinatorial. It immediatelyimplies that the Euler characteristic is independent oftriangulation, because the integral of the Gauss curvatureis. Similarly it implies that the integral of the Gausscurvature remains constant under deformations, becausethe Euler characteristic does.

As you can read in the article on Nirenberg (page126), in 1970 Nirenberg1 asked which functions on thesphere can be the Gauss curvature for some embeddingof the sphere in space. By the Gauss-Bonnet Theorem, theintegral must be 4𝜋. Are there any other conditions?

1Technically Nirenberg considered not only spheres embedded inspace but more abstract Riemannian surfaces given by a Rie-mannian metric on the sphere. But since Nash (who’ll be featuredin the May Notices) proved that any such sphere can be embeddedin high-dimensional space, it really doesn’t matter.

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About the cover: Plain geometry (see page 88)

The Legacy of Vladimir

Andreevich Steklov

page 9

Common Core State

Standards for Geometry: An

Alternative Approach

page 24

Infrastructure: Structure

Inside the Class Group of a

Real Quadratic Field

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Hypergeometric Functions,

How Special Are

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February 2016 Notices of the AMS 145