Announcements:
Class will be held in Kidder 280 MWF at 11 am.
The Course Guide and Calendar contain basic information about Mth 435 for Spring 2014.
Class will be cancelled on Friday April 11, 2014. We will have a make up class on Wednesday April 16 at 2 pm in Kidder 356.
There will also be no classes on Friday April 25, Wednesday April 30 and Friday May 2. Make up classes are scheduled for Wednesday April 23, Wednesday May 7 and Wednesday May 14 at 2 pm in Kidder 356.
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 and is a continuation of Mth 434. We will continue with our study of the geometry of surfaces and will move on to study global theorems about surfaces in Euclidean 3-space. 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.