## 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/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.

## 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.