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21the empire-builder the chern-gauss-bonnet equation

Although it is impossible to summarize a whole century of mathematics in a few sentences, or even in a few pages, some trends can be discerned. The connection between physics and mathematics, which had always been close, in the twentieth century became deeper and more mysterious. Physicists, beginning with Einstein, were routinely startled to discover that mathematicians had already developed the tools they needed. Vice versa, mathematicians kept on discovering that the problems and equations of physics led to the most interesting and deepest mathematics.

Another trend in the twentieth century, connected with the first, was the rise of geometry. Einstein’s theory of general relativity required space to be curved, and this demanded a non-Euclidean geometry whose curvature could vary from point to point. Gauss, Lobachevsky, and Bolyai had sowed the seeds of non-Euclidean geometry in the early 1800s, but their geometries had constant curvature. Bernhard Riemann made it possible to vary the curvature. Riemannian, or differential, geometry developed rather slowly for the first half of the twentieth century, but in the second half it exploded and became a central area of mathematics.

A third important, and invigorating, trend in mathematics was its increasing globalization, especially after the Second World War. Many

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parts of ancient mathematics had been discovered in Asia, Egypt, or the Arab world before they reached western Europe. But from roughly 1500 to 1900, mathematics was mostly a game for European males. Now, with the inequality of opportunity decreasing, the next great discovery is now (almost) as likely to come from a Zhang or an Alice as from a Smith or a Bob.

If I could choose one figure to exemplify all three of these trends, it would be Shiing-Shen Chern. Born in Jiaxing, China (near Shanghai) in 1911, Chern attended Nankai University near Beijing. He distinguished himself enough there to earn a scholarship to study in Europe. He studied for two years in Frankfurt with a geometer named Wilhelm Blaschke, then moved to Paris for a year to work with Elie Cartan.

At that time, differential geometry was not a very fashionable subject. Looking at Einstein’s field equations on page 161 may give you a sense of why. In order to describe the geometry of a curved space (or “manifold”), you need to establish a set of coordinates on it. When equations are written in terms of these coordinates, they are festooned with symbols (like the indices m and n in the field equations) that act merely as bookkeeping devices. Michael Spivak, author of a classic textbook on differential geometry, calls it “the debauch of indices.”

M represents an even-dimensional space or universe with no boundary. χ(M) is the Euler characteristic of the space, which in two dimensions tells you the number of holes it has. Ω is the curvature of the space. The formula allows you to deduce information about the overall shape of the universe if you know its curvature at every point.

∫ P f (Ω) = (2 p ) nχ(M)

M

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Ironically, the most important and interesting quantities in differential geometry are precisely those that do not depend on the choice of coordinates. In other words, we spend all this time keeping track of something that in the end we don’t care about! For example, in Einstein’s theory of general relativity, the independence of physical laws from the coordinate system was a fundamental tenet. Yet it took Einstein years to navigate the mathematics and find equations with the appropriate invariance.

Cartan, Chern’s mentor, had pioneered an approach to differential geometry, called “moving frames,” which worked without coordinates. However, Cartan’s theory was extremely obscure and difficult to understand. Chern became his foremost interpreter for the rest of the world. In the process he transformed Cartan’s theory from a local one, suitable for describing small pieces of a curved space, into a global one that dealt with space as a whole. Chern’s first famous result, considered by many (including Chern himself ) to be his greatest work, was a generalization of a nineteenth-century theorem about surfaces that had been named after Karl Freidrich Gauss and Pierre Ossian Bonnet. The Chern–Gauss–Bonnet theorem, as it is now known, reads as follows:

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studied for two years in Frankfurt with a geometer named Wilhelm Blaschke, then moved to

Paris for a year to work with Elie Cartan.

At that time, differential geometry was not a very fashionable subject. Looking at

Einstein’s field equations on page 000 may give you a sense of why. In order to describe the

geometry of a curved space (or “manifold”), you need to establish a set of coordinates on it.

When equations are written in terms of these coordinates, they are festooned with symbols (like

the indices µ and ν in the field equations) that act merely as bookkeeping devices. Michael

Spivak, author of a classic textbook on differential geometry, calls it “the debauch of indices.”

Ironically, the most important and interesting quantities in differential geometry are

precisely those that do not depend on the choice of coordinates. In other words, we spend all

this time keeping track of something that in the end we don’t care about! For example, in

Einstein’s theory of general relativity, the independence of physical laws from the coordinate

system was a fundamental tenet. Yet it took Einstein years to navigate the mathematics and find

equations with the appropriate invariance.

Cartan, Chern’s mentor, had pioneered an approach to differential geometry, called

“moving frames,” which worked without coordinates. However, Cartan’s theory was extremely

obscure and difficult to understand. Chern became his foremost interpreter for the rest of the

world. In the process he transformed Cartan’s theory from a local one, suitable for describing

small pieces of a curved space, into a global one that dealt with space as a whole.

Chern’s first famous result, considered by many (including Chern himself) to be his

greatest work, was a generalization of a nineteenth-century theorem about surfaces that had

been named after Karl Freidrich Gauss and Pierre Ossian Bonnet. The Chern–Gauss–Bonnet

theorem, as it is now known, reads as follows:

€

Pf (Ω) = (2π )nM∫ χ(M).

What does this mean? From a top-level view, it means that if we live in a curved space or “manifold” (here denoted M), we can learn something about the global shape of our universe (here denoted χ(M), the Euler characteristic of M) by measuring the curvature (Ω) at every point. The Pfaffian (Pf ) is a specific computation we must do with the curvature, and the integral sign (∫) means that we have to add up the curvatures of every point in the manifold. This is a global theorem par excellence.

Let’s burrow a little bit deeper. In this formula the curved space M is assumed to be (2n)-dimensional. So in the simplest case, where n = 1, we are dealing with a two-dimensional space, or a surface. Surfaces have only one local geometric property that is independent of the coordinate frame, called the Gaussian curvature, K. If the surface is convex the Gaussian curvature is positive. If it is shaped like a potato chip, the curvature is negative. The total curvature within any region on the surface is a measure of how much the