Course Description
The course will focus on vector bundles, principal bundles, connections, and curvature, the applications of these concepts in Riemannian geometry, and the relationship between curvature and topology. In the later parts of the course, I am also open to adapting to the desires of the students.
Vector bundles and the associated structures are very natural mathematical objects. We may ask, what is the appropriate notion of "vector-valued functions" on a manifold? If we take the philosophy of manifolds seriously, namely that manifolds are spaces that look locally like Euclidean space but may be different globally, then one answer is "objects that are locally described by vector-valued functions on a chart (but which may have no such description globally)." In the standard terminology this is a section of a vector-bundle. You have met some natural vector bundles in 518: the tangent bundle and the exterior powers of its dual, whose sections are vector fields and differential forms respectively.
Thus vector bundles are fundamental to doing calculus on manifolds. But we need an intrinsic notion of partial differentiation in a vector bundle, which is called a connection. It turns out the basic fact from multivariable calculus
does not always hold in this context. This failure indicates the presence of curvature.
One of the most natural geometric structures on a manifold is a Riemannian metric, a (smoothly-varying) family of inner products on the tangent spaces of the manifold, which lets us measure distances and angles in the manifold. A Riemannian metric determines a connection, whose curvature is then an invariant of the metric up to isometry. It is precisely this curvature which is positive for the sphere, zero for Euclidean space, and negative for the hyperbolic space.
Another theme is the relationship between curvature and
topology. The first result is the Gauss-Bonnet theorem on a
closed 2-manifold 
.
Here 
is the curvature of any Riemannian metric, and 
denotes
the Euler characteristic. There are many generalizations of this
theorem, and many associated ideas such as Chern-Weil theory that we will discuss in the course.
These fundamental concepts provide the foundation for current research in many areas of mathematics, including: Differential Geometry, Riemannian Geometry, and Geometric Analysis (obviously), Algebraic Topology (K-theory), Low-dimensional Topology (Gauge theory), and any subject where manifolds are used.