Instructor | Class meetings | Office hours | Syllabus | References and notes | Exams, homework, and grading | Useful links

Instructor

Class meetings

Tuesday and Thursday 9:30-10:45
Boyd Graduate Studies Research Center, room 326

Office hours

Monday, Wednesday, Friday 12:20-1:10, or by appointment

Syllabus

Topics: Smooth manifolds, smooth maps, tangent vectors, vector bundles, submanifolds, Lie group actions, tensors, differential forms.

The course is designed as preparation for MATH 8260, Differential Geometry II, which will be taught in spring semester 2007 by Sa'ar Hersonsky.

The textbook is Introduction to Smooth Manifolds, by John Lee (Springer 2002). The plan is to cover most of chapters 1-14. The strategy to cover these 387 pages (!) will be to spend more class time on definitions and examples than on proofs. This should work because the book is very readable.

A recommended reference for the end of the course is Riemannian Manifolds: An Introduction to Curvature, by John Lee (Springer 1997). (Note added 10/25: We won't get to any of this material this semester. But this is a good reference for next semester!)

References and notes

A nice description of the three definitions of tangent space can be found in Introduction to Differential Topology, by T. Bröcker and K. Jänich (Cambridge 1982).

Useful facts about flows and differential equations can be found in An Introduction to Differentiable Manifolds and Riemannian Geometry, by William Boothby (second edition, Academic Press 1986).

An informative discussion of flows and vector fields occurs in A Comprehensive Introduction to Differential Geometry, by Michael Spivak, vol. 1, ch. 5 (third edition, Publish or Perish 1999).

An inspiring presentation of Riemann's inaugural lecture, "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen," including a complete translation, occurs in vol. 2, ch. 4 of Spivak's book.

Exams, homework, and grading

There will be no exams. Homework will be assigned every week. Grades will be based on homework and class participation.

Homework 1, due 8/29. Lee p. 28-29: 1-5, 1-6, 1-8.
Homework 2, due 9/5. Lee p. 57-59: 2-6, 2-9, 2-10, 2-11, 2-18.
Homework 3, due 9/12. Lee p. 78-79: 3-2, 3-3, 3-5, 3-6, 3-8.
Homework 4, due 9/19. Lee p. 100-101: 4-2, 4-6, 4-11; p. 460: 17-4; p. 491: 18-1, 18-2.
Homework 5, due 9/26. Lee p. 102: 4-14, 4-16, 4-18; p. 121: 5-2, 5-3, 5-4.
Homework 6, due 10/3. Lee p. 122: 5-11, 5-12; p. 151-152: 6-4, 6-5, 6-6, 6-7.
Homework 7, due 10/10. Lee p. 153-154: 6-11, 6-12; p. 171-172: 7-2, 7-3, 7-5, 7-8.
Homework 8, due 10/17. Lee p. 201-203: 8-1, 8-6, 8-7, 8-10, 8-13, 8-19.
Homework 9, due 10/24. Lee p. 237-238: 9-4, 9-9, 9-10, 9-12, 9-13.
Homework 10, due 11/2. Lee p. 238-240: 9-16, 9-18, 9-22, 9-23, 9-24.
Homework 11, due 11/9. Lee p. 285-290: 11-8, 11-9, 11-10, 11-14, 11-15, 11-18.
Homework 12, due 11/16. Lee p. 122: 5-13, and prove that Λ^k is a smooth functor; p. 319-321: 12-3, 12-4, 12-6, 12-7.
Homework 13, due 11/28. Lee p. 346-348: 13-5, 13-6, 13-8, 13-9, 13-14.
Homework 14, due 12/5. Lee p. 382-384: 14-2, 14-3, 14-6, 14-8, 14-11.

Useful links

Errata for the textbook

This page was created on August 5, 2006. It was last modified on November 29, 2006.