Jordan Watts' Homepage

MATH 6240 - Differential Geometry 2 - Fall 2017

The coaster, the glass, the beer bottle, and the wine bottle are all diffeomorphic as manifolds-with-boundary (ignoring thickness).  However, as Riemannian manifolds (with induced metrics), even after rescaling they all have completely different curvatures.

Welcome to the course webpage! Please read the syllabus, available from the right menu, for policies and more information pertaining to this course.

Course Information

  • Time & Location: MWF 11:00-11:50am in MATH 220.
  • Instructor: Jordan Watts
  • Email: jordan DOT watts AT colorado*
  • Office Hours: M 1:00-1:50pm and 3:00-3:50pm; W 3:00-3:50pm; F 1:00-1:50pm
  • Office: TBA


We will begin by covering Riemannian manifolds; in particular, vector bundles, Riemannian metrics, connections, geodesics, and curvature, with the goal of proving the Gauss-Bonnet Theorem. Next, we will move towards defining spin structures and Dirac operators, covering topics such as principal bundles, heat kernels, maybe Stiefel-Whitney classes? There are a few possibilities, and we will decide which path to take later. The unattainable goal is the Atiyah-Singer Index Theorem; while there probably will not be time to reach it, a sufficient foundation should be created for the interested student to pick up where we left off, with many texts on the subject available.


MATH 6230 (Differential Geometry 1) or equivalent. You should be comfortable with smooth manifolds and smooth maps, tangent bundles, immersions, submersions, smooth embeddings, submanifolds, differential forms, orientations, integration on manifolds, and de Rham cohomology.


Everyone will be expected to give a 20 minute presentation in class (with a 5 minutes questions period afterward) at some point during the course. Your grade will be based on this.

Suggested Texts (none required)

Some of these are available on Springer's website as free ebooks via your CU Boulder Library account.

  • John M. Lee, Riemannian Manifolds - An Introduction to Curvature, Springer, 1997.
    An erratum for the book written by the author can be found here.
  • Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat Kernels and Dirac Operators, 2nd Printing, Springer, 1996.
  • Jürgen Jost, Riemannian Geometry and Geometric Analysis, 5th Edition, Springer, 2008.
    (6th edition may be available)

* Append ".edu", etc., to make this an actual email address. Please do not email me math questions unless they are things that can be discussed easily over email. Especially do not email me with a bunch of symbols in your message. Just come see me after class or during my office hours.