## Math 423: Differentiable Manifolds

### Basic Information

**Instructor: **Eugene Lerman
**e-mail:**lerman@math.uiuc.edu
**Homepage:**
`https://faculty.math.illinois.edu/~lerman`

** Course page:**
`https://faculty.math.illinois.edu/~lerman/423/f02syl.html`

**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

## 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. Vector bundles and operations on
vector bundles.
- 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.
- symplectic and contact structures
- Lagrangian and Hamiltonian formulation of mechanics, Legendre transform.

## Texts

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
K.S. Lam.

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
- An
introduction to differentiable manifolds and Riemannian
geometry by Boothby.

## Grades

The
course grade will be based on weekly homework, one write up
of homework solutions for the class, a midterm and a final.

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Courses Page

Last modified: Thu Aug 29 16:45:16 CDT 2002