Jordan Watts' Homepage

MATH 6230 - Differential Geometry - Spring 2017

Projection of a Klein Bottle into 3-Space (ACME Klein Bottle -- www.kleinbottle.com)

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 12:00-12:50pm in ECCR 116.
  • Instructor: Jordan Watts
  • Email: jordan DOT watts AT colorado*
  • Office Hours: Mondays, 11:00-11:50am (or by appointment)
  • Office: Room 202, Math Building

Curriculum

Time permitting, we will cover as many of the following topics as we can.

  • Smooth Manifolds: implicit function theorem and regular values of functions, coordinate charts on manifolds, topological and smooth manifolds, examples of manifolds: Lie groups, submanifolds defined via implicit functions, and the classification of two-dimensional compact manifolds.
  • Vector Fields and Tangent Bundles: tangent spaces and vectors as differential operators, the tangent bundle over a manifold and triviality, vector fields on manifolds, their local flows, and straightening of vector fields, the Lie bracket of vector fields, as a commutator and as the derivative of a field in the direction of a flow, immersions and submersions.
  • Differential Forms and Tensor Fields: dual spaces and multilinear operators, k-forms on a vector space and wedge products, k-forms and other tensor fields on a manifold, change-of-coordinate formulas for vector fields and k-forms the d operator from k-forms to (k+1)-forms, k-chains and Stokesí theorem on manifolds.
  • Riemannian Manifolds: Riemannian metrics on manifolds, covariant derivatives and the Levi-Civita connection, parallel transport and the geodesic equation, the Riemann curvature tensor and sectional curvature.
  • Other Topics: Whitney's Embedding Theorem, Lie Groups, de Rham cohomology.

Prerequisites

  • MATH 3130 (Linear Algebra) and MATH 4001 (Analysis II), or equivalents. Undergraduates will need my permission to take this course. All students will be expected to be able to write proofs at the level of a mathematics graduate student. If you are not prepared for this, then you should not take this course.

Grades

  • Your grade will be based on seven homework assignments.

Texts

Most of these are available online or I've put them on reserve for the Spring 2017 term in the library. Except for Bredon ... it seems to have gone missing.

  • Main textbook: John M. Lee, Introduction to Smooth Manifolds, 2nd Edition, Springer, 2012
    You can get a free ebook version of this on Springer's Website.
    An erratum for the book written by the author can be found here.
  • Other useful texts include:
    Bredon's Topology and Geometry
    Spivak's Calculus on Manifolds
    Spivak's A Comprehensive Introduction to Differential Geometry, Vol I
    Lee's Introduction to Topological Manifolds, 2nd Edition (especially for the classification of surfaces) - get a free ebook on Springer's Website
    Lee's Riemannian Manifolds: an Introduction to Curvature (especially for the Riemannian Manifold sections)

* 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.