Differential Geometry I and II : MTG 6256-6257
fall 1999-spring 2000
MWF 8th period (3:00-3:50), Matherly 5

Instructor: David Groisser (groisser@math.ufl.edu)

MTG 6256, the first semester of the sequence, will be devoted primarily to the foundations of manifolds. 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; some material will probably be deferred to the second semester, MTG 6257. Please see prerequisites for MTG 6256 below.

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:

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, Spin^c 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

• 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
• Linear algebra, including inner product spaces
• Algebra: basics of groups, rings, homomorphisms, and quotient constructions

Required texts for MTG 6256

Recommended texts (choose either)

Boothby, W., An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, 1986. The library has only the 1975 edition (Dewey# 516.36 B725i), which should not be too different from the newer edition. The 1986 edition lists for $61 at Academic Press and for $58 at amazon.com.

Advantages: This book gives a thorough treatment of the most basic concepts of manifold theory. Many examples are given.

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 (LOC# QA641.S59 1979). Publish or Perish Press seems to have perished despite publishing, so these books may be out of print. Volumes 1 and 2 list for $30 and $25, respectively, 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.

Disadavantages: Too much detail; volume 1 alone is 668 pages long. Also, the fact that these books were photocopied from the typewritten manuscript (rather than typeset) can make for difficult reading.

Some other worthwhile references