Euclidean Geometry is the study of flat space. Between every pair of points there is a unique line segment which is the shortest curve between those two points. These line segments can be extended to lines. Lines are infinitely long in both directions and for every pair of points on the line, the segment of the line between them is the shortest curve that can be drawn between them. Furthermore, if you have a line and a point which isn't on the line, there is a second line running through the point, which is parallel to the first line (never hits it). All of these ideas can be described by drawing on a flat piece of paper. From the laws of Euclidean Geometry, we get the famous theorems like Pythagorus' Theorem and all the formulas you learn in trigonometry, like the law of cosines. In geometry you also learned how to find the circumference and area of a circle.
Now, suppose instead of having a flat piece of paper, you have a curved piece of paper. You might have a cylinder, or a sphere. You can use a cardboard paper towel roll to study a cylinder and a globe to study a sphere. A shortest curve between any pair of points on such a curved surface is called a minimal geodesic. You can find a minimal geodesic between two points by stretching a rubber band between them. The first thing that you will notice is that sometimes there is more than one minimal geodesic between two points. There are many minimal geodesics between the north and south poles of a globe. We can also look for lines, which are curves like the ones in Euclidean space such that between every pair of points on the line, the segment between them is a minimal geodesic. There are no lines on a sphere! Every time you try to extend a minimal geodesic it starts to wrap around and it isn't a minimal geodesic anymore. On a cylinder, some minimal geodesics can be extended to lines but most of them start to wrap around the cylinder and cannot be extended. Surfaces like these are harder to study than flat surfaces but there are still theorems which can be used to estimate the length of the hypotenuse of a triangle, the circumference of a circle and the area inside the circle. These estimates depend on the amount that the surface is curved or bent. One of the basic topics in Riemannian Geometry is the study of curved surfaces.
An important tool used to measure how much a surface is curved is called the sectional curvature or Gauss curvature. It can be computed precisely if you know Vector Calculus and is related to the second partial derivatives of the function used to describe a surface. To study the sectional curvature of a surface at a given point, you first find the tangent plane to the surface at that point. If you can find a small piece of the surface around the given point which only touches the tangent plane at that point, then the surface has positive or zero sectional curvature there. For example, a paraboloid or a sphere has positive sectional curvature at every point. If it is not possible to find a small piece of the surface which fits on one side of the tangent plane, then the surface has negative or zero curvature at the given point. This happens around the neck of a one-sheeted hyperboloid and on points where the surface looks like a saddle. If you use the precise formula to compute the sectional curvature of a point on a plane or a cylinder, then you will discover that these surfaces have exactly zero curvature everywhere.
In Vector Calculus you are also taught how to measure surface area using double integrals. Sometimes when you compute double integrals you use a change of variables and a Jacobian. These techniques are used regularly by Riemannian Geometers.
Riemannian Geometers also study higher dimensional spaces. The universe can be described as a three dimensional space. Near the earth, the universe looks roughly like three dimensional Euclidean space. However, near very heavy stars and black holes, the space is curved and bent. There are pairs of points in the universe which have more than one minimal geodesic between them. The Hubble Telescope has discovered points which have more than one minimal geodesic between them and the point where the telescope is located. This is called gravitational lensing. The amount that space is curved can be estimated by using theorems from Riemannian Geometry and measurements taken by astronomers. Physicists believe that the curvature of space is related to the gravitational field of a star according to a partial differential equation called Einstein's Equation. So using the results from the theorems in Riemannian Geometry they can estimate the mass of the star or black hole which causes the gravitational lensing.
Like most mathematicians, Riemannian Geometers look for theorems even when there are no practical applications. The theorems that can be used to study gravitational lensing are much older than Einstein's Equation and the Hubble telescope. We expect that practical applications of our theorems will be discovered some day in the future. Without having mathematical theorems sitting around for them to apply, physicists would have trouble discovering new theories and describing them. Einstein, for example, studied Riemannian Geometry before he developed his theories. His equation involves a special curvature called Ricci curvature, which was defined first by mathematicians and was very useful for his work. Ricci curvature is a kind of average curvature used in dimensions 3 and up. In Linear Algebra you are taught how to take the trace of a matrix. Ricci curvature is a trace of a matrix made out of sectional curvatures.
One kind of theorem Riemannian Geometers are looking for today is a relationship between the curvature of a space and its shape. For example, there are many different shapes that surfaces can take. They can be cylinders, or spheres or paraboloids or tori, to name a few. A torus is the surface of a bagel and it has a hole in it. You could also stick together two bagels and get a surface with two holes. How many holes can you get? Certainly, as many as you want. If you string together infinitely many bagels then you will get a surface with infinitely many holes in it. Now suppose you make a rule about how the surface is allowed to bend. If a surface must always bend in a rounded way (like a sphere) at every point, then we say it has positive curvature. A paraboloid has positive curvature and so does a sphere. A cylinder doesn't and neither does a torus (look inside the hole to see it bends more like a saddle). There is a theorem which says that if a surface has positive curvature then it cannot have any holes.
A conjecture is a suggestion of a possible theorem which has not yet been proven. In 1969, Milnor stated a conjecture about spaces with positive Ricci curvature. He conjectured that such a space can only have finitely many holes. I am working on trying to find a proof for this conjecture and so are many other Riemannian Geometers. So far there are some partial results. Professors Schoen and Yau showed that 3 dimensional spaces with positive Ricci curvature have no holes at all. On the other hand, Professor Wei has constructed higher dimensional spaces with positive Ricci curvature and many holes, just not infinitely many holes. She doesn't actually build a model with her hands; she describes the spaces explicitly with formulas similar to the way one can describe a globe with an atlas full of maps. The distances between the grid lines are described with formulas and then she does a lot of calculus to compute the Ricci curvature and make sure it is positive. Professors Li, Anderson, and Wilking also have theorems which are related to this conjecture but don't quite prove the conjecture itself. I have proven one theorem which is related to the conjecture. Our theorems can be used as building blocks to find a proof for the whole conjecture but there are still some very important pieces missing. It is almost as if we have put together the outer edge of the puzzle and now we have to fill in the middle. Filling in the middle might be impossible.
Prof. Sormani's research is partially supported by NSF Grant: DMS-0102279.