Instructor: J. Douglas Moore
Textbook: Boothby, An introduction to differentiable manifolds and Riemannian
geometry, 2nd revised edition, 2002, Academic Press, New York. (A copy of the
text will be placed on reserve in the main library. Since I intend to distribute
lecture notes, it may be possible to take the course without buying the text.)
Suggested reading: Spivak, Calculus on manifolds, Harper-Collins, 1965. (A copy of this book will be placed on reserve in the main library.)
Tentative Outline of the Course:
Roughly speaking, differential geometry is the application of ideas from calculus
(or from analysis) to geometry. It has important connections with topology,
partial differential equations and one subject within differential geometry---Riemannian
geometry---is the mathematical foundation for general relativity. Another branch
of differential geometry, connections on fiber bundles, is used in the standard
model for particle physics.
This course will start with calculus on manifolds and will continue to describe the foundations of Riemannian geometry. Further topics may depend on the interests of the participants.
The first part of the course will be devoted to global calculus of several variables, sometimes called "calculus on manifolds." The simplest examples of manifolds are surfaces in Euclidean space---in fact, manifolds are simply the n-dimensional generalizations of surfaces.
Important examples of manifolds are Lie groups, groups which are manifolds in such a way that the group operations are differentiable. Many of the most important groups which arise in geometry and physics (such as the unitary group and the Lorentz group) are Lie groups. Although at first, Lie groups appear to be rather complicated objects which tie together analysis and group theory, there is a remarkable Lie group-Lie algebra correspondence which allows us to reduce many problems in the theory of Lie groups to linear algebra.
One of the key theorems from the global calculus of several variables is Stokes's theorem, a theorem which plays an important role in deRham's beautiful cohomology theory. The de Rham theory provides the most conceptual introduction to ideas from algebraic topology. In fact, Stokes's theorem motivated much of the initial development of algebraic topology, and was used for example in Poincare's 1895 article on Analysis Situs which laid the foundations for modern topology. The de Rham cohomology groups are isomorphic to the real cohomology groups studied in algebraic topology books, and gives these cohomology groups an extremely useful interpretation. We will therefore spend some time developing this theory.
Sometime during the second quarter, we will begin the study of Riemannian geometry and the theory of connections. Riemannian geometry is the study of Riemannian metrics, which determine lengths of curves on a manifold. Any surface in three-dimensional Euclidean space inherits a Riemannian metric, and the Riemannian geometry of surfaces is often studied in undergraduate courses, such as Math 147. We will show how the theory of manifolds makes surface theory far easier to understand. If there is interest among the students we can provide a discussion of many of the topics (minimal surfaces, geodesics on surfaces, the Gauss-Bonnet formula) which occur on the qualifying exam in geometry.
The non-Euclidean geometry discovered by Bolyai and Lobachevsky in the nineteenth century is a special case of Riemannian geometry that has found application in Thurston's program for understanding the topology of three-manifolds. It is helpful to understand well the more general context in which non-Euclidean geometry resides.
Riemannian geometry, with a simple change of sign in part of the metric, is the foundation for general relativity, which makes many amazing predictions, such as existence of black holes.
A basic problem of Riemannian geometry consists of finding the curves of shortest length joining two points in a Riemannian manifold, or more generally, the curves which extremalize length. Such curves are called geodesics. In general relativity, geodesics represent the paths of planets or space ships with their rockets turned off.
Geodesics can be regarded as critical points of an action function on the space of paths joining two points on a Riemannian manifold. Morse developed his beautiful "calculus of variations in the large" to study the relationships between the critical point behaviour of the action and the topology of the space of paths. We intend to discuss some of the beginnings of this theory, which provides a beautiful example of topology applied to geometry through the theory of ordinary differential equations.
The basic local invariant of Riemannian geometry is the curvature, and a major part of the course will be devoted to explaining this concept. In the case of surfaces, the curvature which arises is called the Gaussian curvature. In higher dimensions, many types of curvature have been studied: sectional curvature, Ricci curvature, scalar curvature. Positive sectional curvature implies that nearby geodesics must converge, while negative sectional curvature implies that they diverge. We will discuss the problem of determining relationships between curvature and topology of Riemannian manifolds.
After discussing geodesics and curvature, the remainder of the course will
depend upon student interests. For example, we may head in the direction of
global analysis and discuss Hodge theory, which relates the Laplace operator
on a Riemannian manifold with de Rham cohomology. Or we may go more deeply into
the theory of connections on vector bundles, and discuss Chern classes and similar
topogical invariants. Alternatively, we may give a short mathematical introduction
to general relativity.