What is Geometry?

Geometry is an original field of mathematics, and is indeed the oldest of all sciences, going back at least to the times of Euclid, Pythagoras, and other “natural philosophers” of ancient Greece. Initially, geometry was studied to understand the physical world we live in, and the tradition continues to this day. Witness for example, the spectacular success of Einstein's theory of general relativity, a purely geometric theory that describes gravitation in terms of the curvature of a four-dimensional “spacetime”. However, geometry transcends far beyond physical applications, and it is not unreasonable to say that geometric ideas and methods have always permeated every field of mathematics.

In modern language, the central object of study in geometry is a manifold, which is an object that may have a complicated overall shape, but such that on small scales it “looks like” ordinary space of a certain dimension. For example, a 1-dimensional manifold is an object such that small pieces of it look like a line, although in general it looks like a curve rather than a straight line. A 2-dimensional manifold, on small scales, looks like a (curved) piece of paper – there are two independent directions in which we can move at any point. For example, the surface of the Earth is a 2-dimensional manifold. An n-dimensional manifold likewise looks locally like an ordinary n-dimensional space. This does not necessarily correspond to any notion of “physical space”. As an example, the data of the position and velocity of N particles in a room is described by 6N independent variables, because each particle needs 3 numbers to describe its position and 3 more numbers to describe its velocity. Hence, the “configuration space” of this system is a 6N-dimensional manifold. If for some reason the motion of these particles were not independent but rather constrained in some way, then the configuration space would be a manifold of smaller dimension.

Usually, the set of solutions of a system of partial differential equations has the structure of some high dimensional manifold. Understanding the “geometry” of this manifold often gives new insight into the nature of these solutions, and to the actual phenomenon that is modeled by the differential equations, whether it comes from physics, economics, engineering, or any other quantitative science.

A typical problem in geometry is to “classify” all manifolds of a certain type. That is, we first decide which kinds of manifolds we are interested in, then decide when two such manifolds should basically be considered to be the same, or “equivalent”, and finally try to determine how many inequivalent types of such manifolds exist. For example, we might be interested in studying surfaces (2-dimensional manifolds) that lie inside the usual 3-dimensional space that we can see, and we might decide that two such surfaces are equivalent if one can be “transformed” into the other by translations or rotations. This is the study of the Riemannian geometry of surfaces immersed in 3-space, and was classically the first subfield of “differential geometry”, pioneered by mathematical giants such as Gauss and Riemann in the 1800’s.

Today, there are many different subfields of geometry that are actively studied. Here we describe only a few of them:

1. Riemannian geometry. This is the study of manifolds equipped with the additional structure of a Riemannian metric, which is a rule for measuring lengths of curves and angles between tangent vectors. A Riemannian manifold has curvature, and it is precisely this curvature that makes the laws of classical Euclidean geometry, that we learn in elementary school, to be different. For example, the sum of the interior angles of a “triangle” on a curved Riemannian manifold can be more or less than πif the curvature is positive or negative, respectively.
2. Algebraic geometry. This is the study of algebraic varieties, which are solution sets of systems of polynomial equations. They are sometimes manifolds but also often have “singular points” at which they are not “smooth”. Because they are defined algebraically, there are many more tools available from abstract algebra to study them, and conversely many questions in pure algebra can be understood better by reformulating the problem in terms of algebraic geometry. Moreover, one can study varieties over any field, not just the real or complex numbers.
3. Symplectic geometry. This is the study of manifolds equipped with an additional structure called a symplectic form. A symplectic form is in some sense (that can be made precise) the opposite of a Riemannian metric, and symplectic manifolds exhibit very different behaviour from Riemannian manifolds. For example, a famous theorem of Darboux says that all symplectic manifolds are “locally” the same, although globally they can be extremely different. Such a theorem is far from true in Riemannian geometry. Symplectic manifolds arise naturally in physical systems from classical mechanics, and are called “phases spaces” in physics. This branch of geometry is very topological in nature.
4. Complex geometry. This is the study of manifolds which locally “look like” ordinary n-dimensional spaces that are modeled on the complex numbers rather than the real numbers. Because the analysis of holomorphic (or complex-analytic) functions is much more rigid than the real case (for example not all real smooth functions are real-analytic) there are many fewer “types” of complex manifolds, and there has been more success in (at least partial) classifications. This field is also very closely related to algebraic geometry.

The above list is far from exhaustive. For example, the field of Kaehler geometry is in some sense the study of manifolds which lie in the intersection of the above four subfields.

Finally, another very important area of geometry is the study of connections (and their curvature) on vector bundles, also commonly called “gauge theory”. This field was independently developed by both physicists and mathematicians around the 1950’s. When the two camps finally got together in the 1970’s to communicate, led by renowned figures such as Atiyah, Bott, Singer, and Witten, there resulted a spectacular succession of important new advances in both fields. Some of these accomplishments include the existence of “exotic” 4-dimensional manifolds and the discovery of new invariants that distinguish different types of spaces.

Geometry is more active and exciting than ever, even after 3000 years. And there is no sign of it letting up.