Math 518: Differentiable Manifolds I

### Basic Information

• Instructor: Eugene Lerman
• e-mail: lerman at math uiuc edu
• Homepage: `https://math.uiuc.edu/~lerman`
• Course page: `https://math.uiuc.edu/~lerman/518/f15/518f15.html`
• Office: 336 Illini Hall
• Office Hours: see my calender and/or by appointment
• Phone: 244-9510
• Class meets: MWF 12 am    in 345 Altgeld Hall
• Homework assignments and supplementary notes are posted at
https://math.uiuc.edu/~lerman/518/f15/518f15hw.html

### Prerequisites

Point set topology and linear algebra.

## Course outline

1. Manifolds: Definitions and examples including projective spaces and Lie groups; smooth functions and mappings; submanifolds; Inverse Function Theorem and its applications including transversality; (co)tangent vectors and bundles; vector bundles; manifolds with boundary; orientations.

2. Calculus on Manifolds: Vector fields, flows, and Lie derivative/bracket; differential forms and the exterior algebra of forms; orientations again; exterior derivative, contraction, and Lie derivative of forms; integration and Stokes Theorem, DeRham cohomology.

## Recommended Texts

• Introduction to Smooth Manifolds by John M. Lee, Springer, ISBN: 978-1-4419-9981-8 (Print) 978-1-4419-9982-5 (Online) [free online access from a campus computer]
• Manifolds, Tensors, and Forms An Introduction for Mathematicians and Physicists by Paul Renteln, Cambridge, ISBN: 9781107042193

The course grade will be based on weekly homework (35%), a midterm (25%) and a final (40%).