Math 423: Differentiable Manifolds
Basic Information
- Instructor: Eugene Lerman
- e-mail:lerman@math.uiuc.edu
- Homepage:
http://www.math.uiuc.edu/~lerman
- Course page:
http://www.math.uiuc.edu/~lerman/423/syl.html
- Office: 334 Illini Hall
Phone: 244-9510
- Class meets: MWF 10 am in 159 Altgeld Hall
Prerequisites
Mathematics 323 or 381, or consent of instructor.
If you have any questions or concerns, please contact me by e-mail.
Course outline
The course is an introduction to the language of differentiable
manifolds with a smattering of Riemannian geometry.
We will cover the topics of the differential geometry comprehensive exam:
- Definition and examples of manifolds and of maps of manifolds.
Inverse and implicit function theorem, submanifolds,
immersions and embeddings.
- Tangent vectors, covectors, tangent and cotangent bundles.
- Vector fields, flows, Lie derivative.
- Tensor and exterior algebras. Tensors and differential
forms. Closed and exact forms. Poincare Lemma. de Rham
cohomology.
- Integration of forms on manifolds. Stokes' theorem.
- Riemannian metric, length of curves, geodesics.
- Connections on vector bundles, Levi-Civita connection.
- Curvature of a connection, Riemannian curvature tensor and
sectional curvature.
- Laplace-Beltrami operator, harmonic forms.
Texts
The official text is A comprehensive introduction
to differential geometry, Vol I by M. Spivak
There are three recommended texts:
- Foundations of differentiable manifolds and Lie
groups by F. Warner
- Tensor analysis on manifolds by Bishop
and Goldberg and
- An introduction to differentiable manifolds and
Riemannian geometry
by Boothby.
Grades
The course grade will be based on weekly homework and two exams.
Last modified: Tue Aug 24 13:51:30 CDT