Abstract: The Gauss-Bonnet theorem is an interesting and non-trivial classical theorem. It says that surfaces with certain similar properties will have the same total curvature. In my research project, I deduced corollaries of this theorem.
Imagine an orientable surface with no boundary. Orientable means two-sided; unlike, for instance, a Mobius strip. Examples may be the surface of a basketball, the surface of a very deflated basketball, the surface of a mug, and the surface of a donut (which is called a torus).
Gaussian curvature can be evaluated at each point on such a surface. It measures how curved something is, and whether this curvature is positive or negative. A point in the centre of a saddle, for instance, has negative Gaussian curvature. In one direction, the saddle curves down, but in the other direction, it curves up. The fact that it curves in different directions makes the curvature negative. On the other hand, a point on a sphere has positive Gaussian curvature, as the sphere curves the same way in every direction. A flat plane, and even ‘partially’ curved surfaces like cylinders, have zero Gaussian curvature, as they are flat in at least one direction.
With this in mind, we can consider the total Gaussian curvature of our surface. Here is an example: the total Gaussian curvature of a torus is zero. It turns out that the outside of the torus has positive curvature, the inside has negative curvature, and there happen to be equal amounts on each side. You can also take my word for it that the total Gaussian curvature of our basketball surface is 4π. What about the total Gaussian curvature of the deflated basketball, or the mug?
To answer this, let’s introduce some terminology. Two surfaces are topologically equivalent if one can be continuously deformed into the other, without tearing or puncturing it. So the basketball and the deflated basketball are topologically equivalent, and indeed the deflating of the basketball is a continuous deformation that shows this. Perhaps less obviously, the torus and the mug (assuming the mug has a handle) are topologically equivalent; in this deformation, the hole in the torus becomes the hole formed by the handle of the mug.
The Gauss-Bonnet theorem states that two topologically equivalent surfaces have the same total Gaussian curvature. This is not at all obvious. Two surfaces can be topologically equivalent and yet look very different: they could have different sizes, shapes, distributions of curvature, etc. But by Gauss-Bonnet, the total curvature must be the same.
This tells us that, like the torus, the mug has zero total curvature. Like the inflated basketball, the deflated basketball has 4π total Gaussian curvature. A smooth deformation of a surface will never change its total Gaussian curvature.
In my summer project, I conducted a variation of total Gaussian curvature. That is, I investigated how the total Gaussian curvature would change over time as I smoothly deformed a surface. On one hand, by the Gauss-Bonnet theorem, the total curvature does not change, so that the variation must be zero. On the other hand, I can formally express the variation of Gaussian curvature in terms of different properties of the surface. Concluding that this expression must be zero yields what are called the fundamental equations of submanifold geometry. See my report to discover more!
Annalisa Calvi
Monash University
