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), 79pm at LSB LT2. It will include topics covered up to (and including) the lectures on Oct 22. These include Problem Sets 13 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 Bachelorâ€™s Degrees and Masterâ€™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 201920 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 14 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 14, you receive 20%. If you submit HW 13, and then HW 56, 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 latedrop and Pass/Fail grading. Deadline on Dec 10 (Tue). Apply directly to RES. Please see RES webpage "http://www.res.cuhk.edu.hk/engb/" for detail. Please also see a summary on concerns need to be considered for choosing these options on the departmental website
General Information
Lecturer

LI Manchun Martin
 Office: LSB 236
 Tel: 39431851
 Email:
 Office Hours: by appointment
Teaching Assistant

CHEN Shanjiang
 Office: LSB 222A
 Tel: 39433575
 Email:
 Office Hours: Tue 9am12pm; Wed 9am12pm; Thur 9:30am12pm
Time and Venue
 Lecture: Tue 8:3010:15AM at LSB LT4; Thur 9:3010:15AM at LSB C2
 Tutorial: Thur 8:309:15AM 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 GaussBonnet 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
Preclass Notes
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 arclength, 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, selfadjointness 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
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, 79pm, LSB LT2)  80% 
Useful Links
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.
Assessment Policy Last updated: December 01, 2019 11:05:49