Riemannian Geometry
Riemannian geometry, also called differential geometry, is the study of curved space. It is also the language of general relativity theory (which posits that our universe is a curved space). Finally, it is the most active branch of geometry in contemporary mathematics. Few geometers today study Euclidean circles and triangles and spheres; most of them study manifolds and bundles, which are the basic concepts of Riemannian geometry.
In the 1820s, Carl Friedrich Gauss, Nikolai Lobachevsky, and Janos Bolyai discovered the first non-Euclidean geometry, called the hyperbolic plane. In this geometry, the Euclidean parallel postulate and many of the theorems of Euclidean geometry do not hold; for example, the sum of the angles of a triangle is always less than 180 degrees.
Few mathematicians understood this discovery at first, but the Italian mathematician Eugenio Beltrami, in the 1860s, proved that the hyperbolic plane is no more exotic than a sphere. For example, imagine a triangle drawn on the Earth's surface with one vertex at the North Pole, another on the equator at zero degrees longitude (just off the coast of Africa), and a third vertex on the equator at 90 degrees west longitude (near the Galapagos). With the help of a globe, it is easy to see that this spherical triangle has three right angles, and thus an angle sum of 270 degrees. The excess comes from the positive curvature of the Earth's surface, which makes the lines bulge outward.
Similarly, in the hyperbolic geometry of Gauss, Lobachevsky and Bolyai, the lines of a triangle tend to bulge inward, because the hyperbolic plane is negatively curved. Beltrami even succeeded in drawing a piece of the hyperbolic plane, called a "pseudosphere." Unfortunately, it is impossible to imbed the entire hyperbolic plane in three-dimensional space (a fact proved by David Hilbert). Thus we cannot see it as we can see the surface of a sphere; but we can still imagine it.
In 1854, Bernhard Riemann revolutionized geometry by focusing attention for the first time on space itself, rather than objects in space. The sphere or the hyperbolic plane are very special spaces, because the curvature of space at every point is the same. However, there are many spaces in which the curvature varies from point to point. For example, a torus is positively curved along its outer surface but negatively curved, or saddle-shaped, along its inner surface.
One of Riemann's most important insights was to define the curvature as an intrinsic feature of the space. When we look at a sphere or a torus, we can see its curvature extrinsically, from the way these surfaces bend in the three-dimensional space around them. But Riemann took the point of view of a bug crawling along the surface of the sphere or the torus, which cannot perceive its space from outside. Nevertheless, according to Riemann, it can still observe all the essential geometric features of its space.
The shift from an extrinsic to an intrinsic viewpoint required a whole new conception of what a space is; accordingly, Riemann defined the concept of a manifold, which is still used by mathematicians today. It is a set that can be mapped to a Euclidean space on a small scale, just as cartographers map the globe. Riemann further required the mapping functions to be differentiable (or "smooth"); this allows one to use the tools of calculus, such as derivatives and integrals. Finally, Riemann required the manifold to have an infinitesimal distance scale or metric at every point.
Using these few ingredients, Riemann showed how, from the bug's-eye view, to compute every important geometric feature of a space. He showed how to find geodesics (the equivalent of straight lines), and how to compute distances and volumes inside the manifold. Most importantly, he derived a formula for computing the curvature of a space from its metric alone. For a surface (a two-dimensional manifold), the Riemann curvature at any given point turns out to be a single number. For a three-dimensional manifold the Riemann curvature tensor has 6 components, and for a four-dimensional manifold it has 20. These represent the number of distinct measurements an inhabitant of these spaces would need to make in order to determine what kind of space he or she lives in. Ironically, a curve (a one-dimensional manifold) has no curvature at all. More precisely, the bending of a curve is an extrinsic phenomenon governed by the way it is imbedded in space; it is not detectable by a bug living on the curve.
A great triumph of nineteenth-century differential geometry was the Gauss-Bonnet Theorem, which states that the integral of the Riemann curvature over an entire closed surface (or the "total curvature") is always an integer times 2 pi. The total curvature does not change if the surface is deformed, and is therefore a topological invariant of the surface. It is 4 pi if the surface is a sphere, 0 if it is a torus, and so on. In principle, therefore, an inhabitant of a two-dimensional universe can determine the global structure of his or her universe by integrating local measurements.
Riemann's ideas unexpectedly acquired much greater importance when Albert Einstein, in his theory of general relativity, argued that the universe we live in is curved. Geodesics in this curved space are the paths of light rays; the curvature of space gives rise to gravity. Suddenly, to appreciate the importance of differential geometry, we no longer need to imagine bugs living on surfaces; we are the bugs, and the universe is the four-dimensional manifold we would like to know the shape of. In particular, one solution to Einstein's equations for the shape of space leads to the theory of black holes. Another solution leads to the Big Bang model of cosmology.
A short survey cannot possibly do justice to the many developments in Riemannian geometry in the twentieth century. It is worth mentioning that the century has seen a gradual shift of viewpoint, strongly influenced by physics, toward considering the fundamental object in geometry to be not a manifold but a bundle. Bundles are (roughly speaking) manifolds with a vector space attached at every point. The vector space is a sort of repository for electromagnetic, gravitational, and any other fields.
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