Math 423: Differentiable Manifolds
- Instructor: Eugene Lerman
- Course page:
- Office: 336 Illini Hall
- Office Hours: MW 10 - 10:50 and by appointment.
- Phone: 244-9510
- Class meets: MWF 9 am in 243 Altgeld Hall
Mathematics 323 or 381, or consent of instructor.
If you have any questions or concerns, please contact me by e-mail.
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. Vector bundles and operations on
- Tensors and differential
forms. Closed and exact forms. Poincare Lemma. de Rham
- Integration of forms on manifolds. Stokes' theorem.
- Riemannian metric, length of curves, geodesics.
- Connections on vector bundles, Levi-Civita connection.
- symplectic and contact structures
- Lagrangian and Hamiltonian formulation of mechanics, Legendre transform.
The official text is
Conlon, Lawrence Differentiable manifolds. Second
edition. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser
Boston, Inc., Boston, MA, 2001. xiv+418 pp. $59.95. ISBN 0-8176-4134-3
A recommended text is
Lectures on Differential Geometry by S.S. Chern, W.H. Chen and
Other texts that you may or may not find useful:
A comprehensive introduction
to differential geometry, Vol I by M. Spivak
Other texts you may wish to look at:
- Foundations of differentiable manifolds and Lie
groups by F. Warner
- Tensor analysis on
manifolds by Bishop and Goldberg and
introduction to differentiable manifolds and Riemannian
geometry by Boothby.
course grade will be based on weekly homework, one write up
of homework solutions for the class, a midterm and a final.
Return to Lerman's
Last modified: Thu Aug 29 16:45:16 CDT 2002