## 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.

## 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