Differential Geometry

Differential geometry (Spring 2022) uses techniques from calculus to study geometry, especially curves and surfaces. A central theme in differential geometry is the study of curvature. A small circle seems to be much more curved than a large circle, so we can define the curvature of a circle of radius R to be \frac{1}{R}. For other curves, we can define the curvature at a point to be the reciprocal of the radius of the best-fit, or osculating, circle. This is reminiscent of the idea from calculus that the derivative of a function at a point is the slope of the best-fit, or tangent, line to the graph.

More challenging than the study of curves is the study of surfaces. Here too we can define curvature, but now there are several different types of curvature. There are two principal curvatures, which are the maximum and minimum curvatures at a point. There are then two natural ways of defining a single curvature of a surface at a point: either add or multiply the two principal curvatures, and these two are called mean curvature and Gaussian curvature, respectively.

Gaussian curvature in particular has several remarkable properties. One is that if we transform a surface isometrically, i.e. without changing distances, then the Gaussian curvature also does not change, a fact which is not at all obvious from the definition. This result is known as Gauss’s Theorema Egregium. Another remarkable property of Gaussian curvature is that it determines the topology of the surface. The celebrated Gauss-Bonnet Theorem says that the integral of the Gaussian curvature is related to the Euler characteristic of the surface: \frac{1}{2\pi} \int_M K\, dA=\chi(M).

The mean curvature is also interesting. In particular, surfaces with mean curvature zero are known as minimal surfaces, as they minimize the area of a surface with a fixed boundary curve.

Knowledge of calculus is required for this class.