Handouts

Syllabus

Calendar

Lectures

Week 1: Introduction

Lecture 1: What is a manifold?

Lecture 2: What is a smooth function?

Lecture 3: Tangent vectors

Homework

Problem Set 1 is due Monday, January 10, 2011.

Week 2: Bundles

Lecture 4: The tangent bundle

Lecture 5: Vector bundles

Lecture 6: The cotangent bundle

Homework

Problem Set 2 is due Monday, January 17, 2011.

Week 3: Maps

Lecture 7: Submersions and immersions

Lecture 8: Inverse function theorem

Homework

Problem Set 3 is due Monday, January 24, 2011.

Week 4: Submanifolds

Lecture 9: Embedded submanifolds

Lecture 10: Immersed submanifolds

Lecture 11: Whitney embedding theorem

Homework

Problem Set 4 is due Monday, January 31, 2011.

Week 5: Forms

Lecture 12: Tensors

Lecture 13: Exterior derivative

Lecture 14: Orientations

Homework

Problem Set 5 is due Monday, February 7, 2011.

Week 6: Integration

Lecture 15: Orientations

Lecture 16: Integration

Lecture 17: Stokes' theorem

Homework

Problem Set 6 is due Monday, February 14, 2011.

Week 7: Curves

Lecture 18: Integral curves

Lecture 19: Lie derivatives

Lecture 20: Applications

Week 8: Foliations

Lecture 21: Foliations

Lecture 22: Frobenius theorem

Lecture 23: Poincaré-Hopf index theorem

Week 9: Partial differential relations

Lecture 24: Partial differential relations

Lecture 25: Smale-Hirsch theorem

Lecture 26: h-principle

Week 10: Method of moving frames

Lecture 27: The Maurer-Cartan form

Lecture 28: Curvature via frames