Announcements:
Class will be held in Kidder 280 MWF at 3 pm.
The Course Guide and Calendar contain basic information about Mth 434/534 for Winter 2014.
You can find a practice midterm (from last year) to help prepare for the midterm on February 7, 2014, here.
We will have a make-up class on Monday, March 3, 2014 at 5 pm in Kidder 280.
You can find a practice final (from last year) to help prepare for the second midterm on March 10, 2014, here.
Course Description
A central theme in modern Geometry is the relationship between curvature and topology. The topology of a shape means, roughly, its type up to continuous deformation. For example a round sphere and an ellipsoid (a football) are the same topologically even though they are different geometrically. A torus (the surface of a bagel) is different topologically from the previous two as there is no continuous way of deforming it into the topological type of either one. Thus, given a particular topological shape there are many possible geometric structures that the shape may possess. The round sphere is a very special geometry as it has constant positive curvature, while the surface of a dumb bell, although topologically the same as the sphere, does not have such a symmetric geometry. Its curvature varies from positive to negative. A very important question now arises.
Question: Given a particular topological surface, can we find a geometry with "nice" curvature properties, for example, always positive curvature or always negative curvature, or always 0 curvature? In particular, can we find one with constant curvature?
This course is motivated by the above question. We will begin by studying curvature in the case of 1-dimensional curves in space, before considering their 2-dimensional analogues: surfaces. These will be the topological shapes we will consider. Having discussed curvature from both the intrinsic (a bug living in the surface) and extrinsic (a creature floating above the surface) points of view we will re-consider the above question. In particular we will study one of the most beautiful theorems in Mathematics: the Theorem of Gauss-Bonnet, one of the earliest expositions of the relationship between topology and geometry.