Instructor: David Groisser (groisser@math.edu)
Course summary. This year-long graduate sequence introduces the tools of differential geometry and differential topology.MTG 6256, the first semester of the sequence, will be devoted primarily to the foundations of manifold theory. Topics will include a brief review of advanced calculus from a geometric viewpoint; definition and examples of manifolds (including Lie groups and submanifolds of Euclidean space); maps of manifolds; critical points, the Regular Value Theorem, and Sard's Theorem (possibly deferred to spring); vector fields, flows, and Lie derivatives; exterior algebra and differential forms; Stokes' theorem and de Rham cohomology; vector bundles and tensor bundles; Riemannian metrics; introduction to Riemannian geometry; and connections and curvature. Riemannian geometry and the theory of connections are large topics; most of this material will probably be deferred to the second semester, MTG 6257. Please see prerequisites for MTG 6256 below. Here is a syllabus.
MTG 6257. If time forbids covering the following topics in the fall semester (MTG 6256), they will likely be covered in the spring (MTG 6257): surfaces in Euclidean space and the Gauss-Bonnet theorem; connections on principal bundles and associated vector bundles. Some possibilities for additional topics for the spring semester are:
There will be time only for a very limited number of these topics (some of them are semester-long topics by themselves). Student input will be sought before a final decision is made.
- Further study in Riemannian geometry (conjugate points on geodesics, Hopf-Rinow Theorem, Morse index, curvature comparison theorems, ...)
Lie groups and Lie algebras
Elliptic PDE on manifolds and Hodge Theory
Curvature and characteristic classes
Gauge theory and the Yang-Mills equations
Spin bundles, Spinc bundles, Dirac operators, and the Seiberg-Witten equations
Symplectic geometry and the geometry of classical mechanics
Complex and Kaehler manifolds
Selected topics in differential topology (transversality, Poincare-Hopf Theorem, degree theory, embedding theorems, ...)
Prerequisites for MTG 6256
Essentially, these can be summarized as "Everything you would learn as an undergraduate (pure-)math major". Differential geometry pulls together strands of all three major divisions of mathematics: analysis, algebra, and topology. Here are some particulars:
- Linear algebra, including inner-product spaces.
This is essential. By linear algebra I don't just mean matrix algebra (although the basics of matrix algebra are still essential) or numerical linear algebra (working with large linear systems on a computer); I mean the general concepts and theory of vector spaces, linear transformations, and inner products. Differential geometry uses linear algebra probably more than any other field does, and it's important that you be comfortable with it. By the end of the course you will know a lot more about linear algebra than when you started, but you have to start at a reasonable level. If you are not clear on the difference between a matrix and a linear transformation, you are not ready to learn differential geometry.
- Advanced calculus. Although the topics below will be reviewed briefly, students entering the class should already be familiar with:
- the derivative viewed as a linear transformation
- the chain rule
- the Inverse Function Theorem and Implicit Function Theorem
- the fundamental existence/uniqueness theorem for (systems of) ordinary differential equations
- Basic point-set topology
- metric spaces (familiarity with more general topological spaces would be helpful as well)
- open and closed sets; compactness; connectedness; completeness
- continuous maps and homeomorphisms
These topics are reviewed briefly in this Point-Set Topology: Glossary and Review, which students should go through before the course starts.- Algebra: basics of groups, rings, homomorphisms, and quotient constructions. I hesitate to call all of this a prerequisite, since much of it will be used only occasionally. However, when it is used, if you're unfamiliar with the foundational material I'm using, it will be up to you to learn what you need on your own.
Required texts for MTG 6256
None, since I will not be following any textbook very closely for more than a few weeks at a time. However, to supplement the lectures I recommend that students obtain either Boothby or "Big Spivak", listed below. Both are (or will soon be) on reserve at the Marston Science Library.Recommended texts (choose either)
- Boothby, W., An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., Academic Press, 1986. The edition that will be on reserve is the 1975 edition (QA3.P8 v.63), since that's the only one the library has. At the time this webpage is being written, amazon.com lists the 1986 edition at various prices from $49.90 to $88.80.
Advantages: This book gives a thorough treatment of the most basic concepts of manifold theory, and a good review of the relevant prerequisites from advanced calculus. Many examples are given. I have a copy on my shelf, so it will be easy for you to ask me questions about something you read. I will be consulting it occasionally when preparing my lectures.
Disadvantages: For the level and topics I intend for this course, the ratio of advanced to elementary material is too small.
("Big Spivak") Spivak, M., A Comprehensive Introduction to Differential Geometry, volumes 1 and 2. Publish or Perish, 1979 (QA641.S59 1979). Publish or Perish Press seems to have perished despite publishing, so these books may be out of print, but you can still find new and used copies at amazon.com.
Advantages: Spivak's encyclopedic five-volume series is very down-to-earth. More motivation and historical development is given here than in any other text I know.
Disadvantages: Too much detail; volume 1 alone is 668 pages. My belief is that one learns better if more is left to the reader. Also, the fact that these books were photocopied from the typewritten manuscript (rather than typeset) can make for difficult reading. I own only volumes 4 and 5, so I may not be easily able to answer questions about something you read in volumes 1 and 2. I will not be consulting Spivak when preparing my lectures.