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...