Math 562: Introduction to Differential Geometry and Topology
Professor: Kiril Datchev
Lectures: Mondays, Wednesdays, and Fridays, 12:30 to 1:20, in UNIV 101.
Office hours: Wednesdays 1:30 to 2:30, Thursdays 2:00 to 3:30, or by appointment, in MATH 602.
Textbook: An Introduction to Differentiable Manifolds and Riemannian Geometry, by William M. Boothby. The link is for the second edition which is fine, but the revised second edition is slightly better. (The first edition also fine but is missing some homework problems we want to use.) For more on differential forms we are also using Lorenzo Sadun's lecture notes and Manfredo P. do Carmo's Differential Forms and Applications.
We will cover the following topics: smooth manifolds; tangent vectors; inverse and implicit function theorems; submanifolds; vector fields; integral curves; differential forms; the exterior derivative; DeRham cohomology groups; surfaces in Euclidean Space, Gaussian curvature; two-dimensional Riemannian geometry; Gauss–Bonnet and Poincaré theorems on vector fields.
Grading is based on
Almost weekly homework assignments, worth 1/3 of the total grade,
two in-class midterm exams, on dates to be announced, worth 1/3 of the total grade,
a final exam, as scheduled here, worth 1/3 of the total grade.
Homework is due on paper at the beginning of class on Fridays. Here are the assignments:
Homework 1, due September 2nd. See also this note on the definition of a manifold.
Homework 2, due September 9th. See also this note on the inverse function theorem.
Homework 3, due September 16th.
The first midterm will be in class on Monday September 26th.
Homework 4, due October 7th. See also this note on ordinary differential equations.
Homework 5, due October 14th.
Homework 6, due October 21st. For a concrete example of a partition of unity, see Figure 16.4 of Munkres' book
The second midterm will be in class on Monday October 31st. See also this note on change of variables.
Homework 7, due November 11th.
Homework 8, due November 18th.
Homework 9, due December 2nd.
The final exam will be on December 13th, as scheduled here.
Below are some books recommended for further reading.
Much of the material in the course can be found in Michael E. Taylor's'
Introduction to Analysis in Several Variables, also available as a preprint.
Topology from the Differentiable Viewpoint, by John W. Milnor, and Differential Topology, by Victor Guillemin and Alan Pollack, are beautiful, classic treatments of differential topology.
If you like Milnor's book, his more challenging Morse Theory introduces Riemannian geometry.
A longer and more detailed introduction to everything in our course and more can be found in John M. Lee's two books Introduction to Smooth Manifolds and Introduction to Riemannian Manifolds.
Visual Differential Geometry and Forms, by Tristan Needham, has lots of attractive and illuminating figures, and ample discussions of the history of the subject and its relationship to physics.
Finally, a list of general policies and procedures can be found here.