Erlanger Program

Erlanger program-an article written by the great mathematician Felix Klein in 1872, in which he outlined his vision of the direction in which the study of geometry should proceed in the years that followed. Klein's key idea was that geometric structures on spaces should be understood in terms of the group of transformations that preserve the structures. Klein's insight, which seemed very abstract to his contemporaries, was a source of inspiration to the mathematicians that followed him, notably Henri Poincaré, and has become fundamental to our current understanding of geometry.

In the years preceding Klein's Erlanger program, the world of geometry had been changed by the discovery of non-Euclidean geometries, in which Klein was instrumental. Mathematicians realized that the same space could admit more than one geometric structure, and it became important to make a precise definition of a geometric structure on a space. Two complementary pictures evolved. One was the idea of Riemann that a geometric structure was given by a local definition of lengths and angles; this gave rise to the field of differential geometry. The other was Klein's Erlanger program, which argued that a geometric structure could be defined in terms of the transformations that left it invariant.

A transformation of a space is a mapping of the space onto itself (usually required to be continuous and have a continuous inverse) that never maps two points to the same point. For example, in the plane, a rotation (say, a quarter-turn) about a fixed point is a transformation, since it maps the plane onto itself, and no two different points get rotated to the same point. Another transformation is the 'stretch' map that leaves the origin fixed, and sends every other point to the point twice as far from the origin, along the same line (in symbols, this would be the function f(x,y)=(2x,2y)). Like the rotation, the stretch map sends the plane onto itself, and no two points get mapped to the same point. On the other hand, a map that sends the plane onto the upper-half plane by folding it along the x-axis like a sheet of paper is not a transformation, since it does not map the plane onto the entire plane, and two different points can get mapped to the same point.

A transformation is said to preserve a geometric structure if it leaves all the measurements of that structure (lengths, angles, etc.) unchanged. Thus, the rotation preserves the Euclidean structure on the plane, since rotation does not change angle or length measurements. On the other hand, the stretch map does not preserve the Euclidean structure on the plane; it preserves angle measurements, but not length measurements.

According to Klein, a geometric structure consists of a space together with a particular group of transformations of the space. The transformations that preserve Euclidean space are the rotations, translations (maps that shift the plane in some fixed direction, by some fixed amount), reflections across a line, and glide reflections (maps that consist of a reflection across a line, followed by a shift in the direction of the line). Klein would say that two-dimensional Euclidean geometry is the two-dimensional plane together with the group of rotations, translations, reflections and glide reflections of the plane. Hyperbolic geometry can be modeled by the upper-half plane, together with the following transformations: horizontal translations, reflections across vertical lines, stretch maps with center on the x-axis, inversions with center on the x-axis, and combinations of those maps (an inversion is a map that leaves a given circle fixed, and sends every other point to a point on the same line through the center, according to a precisely defined rule).

As mathematicians began to have a better understanding of what constitutes a geometric structure, they realized that the coordinate line, plane and higher-dimensional spaces were far from the only spaces that could admit geometric structures. The notion of a manifold evolved: a manifold is a space for which, around every point, there is a neighborhood that looks like a bent or distorted piece of the coordinate space of the appropriate dimension. Thus, the surface of the earth is a 2-dimensional manifold, since every point has a small neighborhood that looks like a slightly curved piece of the coordinate plane.

Riemann's and Klein's ideas generalize to give definitions of what it means to have a geometric structure on a manifold. Riemann would say that a geometric structure consists of a definition of lengths and angles on the manifold. But a geometric structure can also be defined in terms of transformations. When we say that every point in a manifold has a neighborhood that "looks like" coordinate space, what we mean is that there is a function that maps the neighborhood onto a piece of coordinate space. But there is not just one such function; each point has many neighborhoods that look like many different pieces of coordinate space. Hence there are many different pieces of coordinate space associated to the same piece of the manifold, and there are transformations that take one of the corresponding pieces of coordinate space to another; these transformations are called the transition maps. We can put a geometric structure on the manifold by requiring that the transition maps satisfy geometric requirements. A manifold is said to have a Euclidean structure if the transition maps preserve Euclidean length and angle measurements. A manifold has a hyperbolic structure if the transition maps preserve hyperbolic measurements. This type of definition is not limited to geometric structures: a manifold is said to have a differentiable structure if the transition maps and their inverses are differentiable. The question of what kinds of additional structures different manifolds can support is one of the most active fields of mathematical research today.