Differentiable Manifolds, Fall 2021

Lecturer:

  • Prof. Raphael Ponge.
  • E-mail: ponge[dot]math[at]icloud[dot]com.

Time and Location:

  • Tuesdays and Thursdays, from 8:00-8:45 & 8:55-9:40, 3rd Teaching Building, Room 215.
  • Until I can come back to China the lectures will be online. Here is the Zoom link. Meeting ID: 978-2997-3788. Passcode: 5KLaef.
  • The lectures are from Week 1 through Week 12. The last class is on November 18, 2021.

Main Reference:

  • An Introduction to Manifolds by Loring W. Tu, 2nd edition, Universitext, Springer, 2011 (Springer Link; free access from SCU campus).
  • Click here and use the password provided during the 1st lecture to get an e-copy of the book. Please send me an e-mail if you haven’t got the password.

Further Reading:

  • Lectures on Differential Geometry by S.S. Chern and Wei-Huan Chen, Peking University Press, 1983. English translation published by World Scientific Press, 1999.
  • Introduction to Smooth Manifolds by J.M. Lee, Springer, 2nd edition, 2014.

Grading Scheme:

  • Your final grade will be based on your scores on the homework assignments and the Final Exam.
  • There will be 5 take-home homework assignments.
  • The date of the Final Exam will be announced later.

Homework Assignment:

  • Homework 1 (due September 16):
    • Problems #1.4, 2.4, 3.4, 3.5, 3.8, 4.6.
    • For #3.4 look at transpositions and use Lemma 3.11.
    • For #3.5 use #3.4.
    Note: The homework papers should be handed in to Prof. Xiaojun Chen in Room East 420 (School of Mathematics) between 14:00 and 17:00 on September 16.

  • Homework 2 (due September 30):
    • Problems # 5.5, 6.2, 6.4, 7.6, 7.7, 8.2, 8.4.
    • For #7.6, 7.7 show that the quotient space is homeomorphic to the unit circle.

  • Homework 3 (due October 21):
    • Problems # 9.7, 9.8, 10.4, 10.5, 11.2, 12.3.

Tentative Schedule:

  • Aug. 31: Section 1 (slides) + Section 2 (slides). Recommended problems: #1.2, 1.3, 1.5, 1.6, 1.8, 2.1, 2.2, 2.3.
  • Sep. 2: Section 3 (slides). Recommended problems: #3.7, 3.9, 3.10, 3.11.
  • Sep. 7: Section 4 (slides). Recommended problems: #4.1, 4.2, 4.3, 4.4.
  • Sep. 9: Section 5 (slides). Recommended problems: #5.1, 5.2, 5.3, 5.4.
  • Sep. 14: Section 6 (slides). Recommended problems: # 6.1, 6.3.
  • Sep. 16: Section 7 (slides). Recommended problems: # 7.1, 7.2, 7.3, 7.4, 7.5, 7.8, 7.9.
  • Sep. 21: No classes, Mid-Autumn Festival.
  • Sep. 23: Section 8 (slides). Recommended problems: # 8.1, 8.3, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10.
  • Sep. 28: Section 9 + Section 10. Recommended problems: # 9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 9.10, 10.4, 10.5, 10.7.
  • Sep. 30: Section 11. Recommended Problems: # , 11.1, 11.2, 11.4, 11.5.
  • Oct. 5 & 7: No classes, National Day Holiday.
  • Oct. 12: Section 12. Recommended problems #12.1, 12.2, 12.3, 12.4.