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WHATIS... Gauss Curvature? Editors

Gaussian curvature is a curvature intrinsic to a two- Take the curvature of the slice-curve. Thus you get a dimensional surface, something you’d never expect a curvature in every direction. The largest (positive, upward) surface to have. A bug living inside a curve cannot tell if and smallest (most negative) such curvatures are called it is curved or not; all the bug can do is walk forward and the principal curvatures, and they occur in orthogonal backward, measuring distance. But a very intelligent bug directions. Their average is called the mean curvature, living on a surface can, by walking around, never leaving and their product is called the Gauss curvature. For the the surface, measure its Gaussian curvature. unit sphere, both principal curvatures are 1 and hence The Definition. Assuming you know how to define the the Gauss curvature is 1. For a unit cylinder, the principal curvature of a curve, how would you define the curvature curvatures are 1 and 0 and hence the Gauss curvature of a surface at a point 푝? Slice the surface by a plane is 0. For a suitable hyperbolic paraboloid as in Figure 2, normal to the surface at 푝, as in Figure 1. the principal curvatures are 1 and −1, and the Gauss curvature is −1. , Figure 3.1. Riemannian Geometry Image courtesy of Cesar E. Silva.

Image by J. Bredt; ©Frank Morgan, Figure 2. For this hyperbolic paraboloid, the Gauss Figure 1. To define the Gauss curvature, consider the curvature is (+1)(−1) = −1. curvature of slice curves. The Gauss curvature is the product of the largest (positive) and the smallest or Gauss’s Theorema Egregium says that this Gauss cur- most negative such curvatures. vature is intrinsic; in other words, it can be measured by a bug within the surface. That means that if you bend the surface without stretching it, the Gauss curvature cannot For permission to reprint this article, please contact: change. If you roll the plane up into a cylinder, as in [email protected]. Figure 3, the principal curvatures change from 0, 0 to 0, 1, DOI: http://dx.doi.org/10.1090/noti1333 but the Gauss curvature remains 0.

144 Notices of the AMS Volume 63, Number 2 THEGRADUATESTUDENTSECTION

As you can read in the article on Nirenberg (page 126), in 1970 Nirenberg1 asked which functions on the sphere can be the Gauss curvature for some embedding of the sphere in space. By the Gauss-Bonnet Theorem, the integral must be 4휋. Are there any other conditions? Riemannian

1Technically Nirenberg considered not only spheres embedded in space but more abstract Riemannian surfaces given by a Rie- mannian metric on the sphere. But since Nash (who’ll be featured in the May Notices) proved that any such sphere can be embedded , Figure 3.7. in high-dimensional space, it really doesn’t matter. Image by J. Bredt;Geometry ©Frank Morgan, Figure 3. Rolling a piece of plane into a cylinder of radius 푟 changes the principal curvatures from 0, 0 to 1/푟, 0 but the Gauss curvature–the product–remains 0.

One way a bug can measure the Gauss curvature of a surface at a point 푝 is by looking at a tiny “geodesic” circle of tiny radius 푟 about 푝: {푥 ∶ dist(푝, 푥) = 푟}. If the Gauss curvature is 0, the circumference is 2휋푟 to high order. If the Gauss curvature is positive as on the sphere, the circumference is smaller. If the Gauss curvature is negative as on the hyperbolic paraboloid, the circumference is greater. In general, for Gauss curvature 퐺, the circumference is given by 푟3 2휋푟 − 퐺휋 + … . Moving? 3 For a closed surface, there is the amazing Gauss- Please make sure that the AMS Notices Bonnet formula relating the Gauss curvature 퐺 and and Bulletin find their new home. the Euler characteristic 휒. The Euler characteristic is a ISSN 0002-9920 (print) ISSN 1088-9477 (online) purely combinatorial quantity, easily computed from any

Volume 61, Number 1 of the American Mathematical Society triangulation as the number of vertices minus the number January 2014 The Legacy of Vladimir Andreevich Steklov page 9

Common Core State Standards for Geometry: An Alternative Approach of edges plus the number of regions. The Gauss-Bonnet page 24 Infrastructure: Structure Inside the Class Group of a Real Quadratic Field page 36

Hypergeometric Functions, How Special Are They? Theorem says that page 48

Knoxville Meeting page 100

Baltimore Meeting page 103

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Lubbock Meeting page 111

Tel Aviv Meeting ∫ 퐺 = 2휋휒. page 114 About the cover: Plain geometry (see page 88)

Since the Euler characteristic of the sphere is 2, this says • that the integral of the Gaussian curvature is 4휋. This is Email your new address to us: easy to check for the round unit sphere: since the Gauss [email protected] • curvature is 1, it just says that the area is 4휋. The amazing or make the change yourself online at: thing is that it remains true for a deformed sphere of any www.ams.org/cml-update size and shape. • or send the information to: The Gauss-Bonnet Theorem is an amazing equality Member and Customer Services of the geometric and the combinatorial. It immediately American Mathematical Society implies that the Euler characteristic is independent of 201 Charles Street triangulation, because the integral of the Gauss curvature Providence, RI 02904-2294 USA is. Similarly it implies that the integral of the Gauss Phone: (800) 321-4267 (US & Canada) curvature remains constant under deformations, because (401) 455-4000 (Worldwide) the Euler characteristic does.

February 2016 Notices of the AMS 145