MATH4030 - Differential Geometry - 2019/20

Announcement

• There is no tutorial in the first week.
• Lecture notes from previous years are available under "Useful Links". Also, a brief notes for preliminary materials is posted.
• Class schedule changes: we will have a double lecture on Oct 17 and a double tutorial on Oct 24.
• The midterm will take place on Oct 30 (Wed), 7-9pm at LSB LT2. It will include topics covered up to (and including) the lectures on Oct 22. These include Problem Sets 1-3 and some of Problem Set 4. You can find sample midterms from previous years under the "Quizzes and Exams" section.
• The numerical answers to Q1 of Problem Set 4 for S_1 is given by K=4/(1+4x^2+4y^2)^2 and H=4(1+2x^2+2y^2)/(1+4x^2+4y^2)^3/2. For Q2, H=0 and K=-sech^4 v.
• Note that there is no lecture or tutorial on Nov 7 due to the Conferment of Bachelor’s Degrees and Master’s Degrees.
• The midterm has been returned. The mean is 49, median is 53, with SD 13. You can collect the midterm from me at my office LSB 236 or after class next week.
• Since the semester has ended early due to circumstances happening at CUHK, we will make announcement later about the arrangement for course assessment etc. at a later time.
• Due to the shortening of Term 1 in 2019-20 and cancellation of the centralized final examination, we will use the following new assessment scheme for this course: Homework (20%) and Midterm (80%). Only Homework 1-4 will be counted towards the course grade.
• You are still welcomed to submit HW 5 and HW 6 by email to me or the TA. The deadline for any homework submission is Dec 6, 2019 at 23:59pm. If you choose to submit HW 5/6, your scores toward Homework (20%) will be scaled accordingly. For example, if you submit all HW 1-4, you receive 20%. If you submit HW 1-3, and then HW 5-6, you receive 16.7%.
• New update: For the homework assessment, only the 4 best homework out of the total of 6 will be counted.
• University has just announced the options for late-drop and Pass/Fail grading. Deadline on Dec 10 (Tue). Apply directly to RES. Please see RES webpage "http://www.res.cuhk.edu.hk/en-gb/" for detail. Please also see a summary on concerns need to be considered for choosing these options on the departmental website

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General Information

Lecturer

• LI Man-chun Martin
  • Office: LSB 236
  • Tel: 3943-1851
  • Email:
  • Office Hours: by appointment

Teaching Assistant

• CHEN Shanjiang
  • Office: LSB 222A
  • Tel: 3943-3575
  • Email:
  • Office Hours: Tue 9am-12pm; Wed 9am-12pm; Thur 9:30am-12pm

Time and Venue

• Lecture: Tue 8:30-10:15AM at LSB LT4; Thur 9:30-10:15AM at LSB C2
• Tutorial: Thur 8:30-9:15AM at LSB C2

Course Description

This course covers basic theory on curves, and surfaces in the Euclidean three space. Topics include: regular curves, Frenet formulas, local theory of curves, global properties of curves such as isoperimetric inequality, regular surfaces, 1st and 2nd fundamental form, Gaussian curvature and mean curvature, Gauss map, special surfaces such as ruled surfaces, surfaces of revolution, minimal surfaces, intrinsic geometry: geodesic, and Gauss-Bonnet Theorem. Students taking this course are expected to have knowledge in advanced calculus, linear algebra, and elementary differential equations.

Textbooks

• "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo, Prentice Hall, 1976

Pre-class Notes

• Course outline
• Preliminary materials

Class Notes

• Sep 3: Course outline, brief overview of the course
• Sep 5: Definition of curves with examples, arc length
• Sep 10: properties of arc length, regular curve, reparametrization by arc-length, curvature and Frenet frame for plane curves
• Sep 12: Frenet frame for space curves and examples
• Sep 17: Fundamental theorem of plane/space curves, simple closed curves, theorem of turning tangent
• Sep 19: Isoperimetric inequality
• Sep 24: Definition of surfaces, examples including graphs, surfaces of revolution and ruled surfaces
• Sep 26: Inverse/Implicit Functions Theorems, level surfaces
• Oct 3: Change of parameters, smooth functions on surfaces
• Oct 8: Tangent planes and differentials
• Oct 10: Integration on surfaces, vector fields on surfaces, orientability
• Oct 15: first fundamental form, gauss map, shape operator, Gauss and mean curvatures
• Oct 17: second fundamental form, self-adjointness of shape operator, gauss/mean curvature in local coordinates
• Oct 22: local expressions of Gauss/mean curvatures, normal curvatures, principal curvatures and directions, totally umbilic surfaces
• Oct 29: intrinsic geometry of surfaces, isometry and local isometry, vector fields as directional derivatives
• Oct 31: Lie bracket of vector fields, differentiating vector fields in R^n
• Nov 5: Covariant derivative, Einstein summation convention, Christoffel symbols
• Gauss and Codazzi equations (do Carmo 4.3)
• Parallel Transport and Geodesics (do Carmo 4.4)
• Gauss Bonnet Theorems (do Carmo 4.5)

Tutorial Notes

• Tutorial notes 1 (Sep 12)
• Tutorial notes 2 (Sep 19)
• Tutorial notes 3 (Sep 26)
• Tutorial notes 4 (Oct 3)
• Tutorial notes 5 (Oct 10)
• Tutorial notes 6 (Oct 24)
• Tutorial notes 7 (Oct 31)

Assignments

• Problem Set 1 (due on Sep 19)
• Problem Set 2 (due on Oct 4)
• Problem Set 3 (due on Oct 22)
• Problem Set 4 (due on Nov 5)
• Problem Set 5 (due on Nov 19)
• Problem Set 6

Quizzes and Exams

• Midterm (2016-17 Term 1)
• Midterm (2017-18 Term 1)

Solutions

• Solution to Problem Set 1
• Solution to Problem Set 2
• Solution to Problem Set 3
• Solution to Problem Set 4
• Solution to Problem Set 5
• Solution to Problem Set 6
• Solution to Midterm

Assessment Scheme

Homework		20%
Midterm (Oct 30, 7-9pm, LSB LT2)		80%

Useful Links

• Lecture notes from 2017
• Lecture notes from 2015

Honesty in Academic Work

The Chinese University of Hong Kong places very high importance on honesty in academic work submitted by students, and adopts a policy of zero tolerance on cheating and plagiarism. Any related offence will lead to disciplinary action including termination of studies at the University. Although cases of cheating or plagiarism are rare at the University, everyone should make himself / herself familiar with the content of the following website:

http://www.cuhk.edu.hk/policy/academichonesty/

and thereby help avoid any practice that would not be acceptable.

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Assessment Policy

Last updated: December 01, 2019 11:05:49