## Math 423: Differentiable Manifolds

### Basic Information

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

** Course page:**
`https://faculty.math.illinois.edu/~lerman/423/f03syl.html`

**Office:** 336 Illini Hall
**Office Hours:** MW 10 - 10:50 and by appointment.
**Phone:** 244-9510
**Class meets:** MWF 9 am in 441 Altgeld Hall

## Prerequisites

Mathematics 323 or 381, or consent of
instructor. Point set topology and linear algebra will be very
useful.
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.
- Curvature
- Lagrangian and Hamiltonian formulation of mechanics, Legendre transform.

## Texts

Here are lecture notes of the
last year's version of the course taken by David Rose (pdf format).

Recommended texts are

An introduction to Riemannian manifolds and differential geometry
by Boothby

Foundation of differentiable manifolds and Lie groups
by Warner.

Other texts that you may find useful:

A comprehensive introduction
to differential geometry, Vol I by M. Spivak

Tensor analysis on
manifolds by Bishop and Goldberg
## Grades

The
course grade will be based on weekly homework,
a midterm and a final.

Return to Lerman's
Courses Page

Last modified: Thu Sep 18 12:40:17 CDT 2003