Introduction to Differential Geometry: MATH 4540
Spring 2019

Instructor:   Xiaodong Cao, Office:521 Malott, Phone: 5-7431

E-mail: cao at math.cornell.edu

Teaching Assistant: Beihui Yuan (Email: by238 at cornell.edu OH: Tuesday 2-4 pm, 218 MLT)

Time and Place:  10:10 am - 11:25 am, Tuesday and Thursday, 203 MLT

Office Hour: Thursday 2:00-2:50 pm or by appointment.

Text:  The following book is recommended:

Prerequisites: Calculus and Linear Algebra. The course will be self contained. But a little bit knowledge of Differential Equation and Topology will also help.

Midterm Exams:  The midterm exams are on March 7th and April 16th.   Make-ups will not be given for the midterm exams. Students can only be excused from the midterms because of serious illness or a family emergency of comparable gravity. To be excused you will need a note from your doctor or dean.

Final Exam: The final exam (take-home) is due on May 16th.

Homework: Homework will be assigned every two week and will be due on the date stated on the homework. The homework assignments will be announced in class. You must hand in the homework at the beginning of class each Tuesday. Late homework will NOT be accepted under any circumstances.

Grading: The course grade is apportioned as follows: Final exam 30%; the first midterm exam 25%; the second midterm 25%; homework grades 20%.

Academic honesty: It is the obligation of each student to understand the Cornell Code of Academic Integrity regarding academic honesty and to uphold these standards. This states, "A Cornell student's submission of work for academic credit indicates that the work s the student's own. All outside assistance should be acknowledged, and the student's academic position truthfully reported at all times." Students are encouraged to talk about the problems, but should write up the solutions individually. Students should acknowledge the assistance of any book, software, student or professor.

Copyright : Course materials posted on this website or distributed in class are intellectual property belonging to the author. Students are not permitted to buy or sell any course materials without the express permission of the instructor. Such unauthorized behavior constitutes academic misconduct.

Disabilities: Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see the instructor as soon as possible. Also, stop by the Office of Disability Services to register for support services.


 

Schedule of Lectures (Tentative!)

Class Topic Read   Exercises Due
Jan. 22 Introduction Ch 1 HW 1 2/7
Jan. 24 Point-set topology  Ch 2    
Jan. 29 Maps Ch 3    
Jan. 31 Derivative of maps   HW 2 2/21
Feb. 5 Composition of maps Space curves, curvature    
Feb. 7 Proper maps, inverse function theorem      
Feb. 12 Properties and examples      
Feb. 14 Inverse function theorem, implicit function theorem, parametric function theorem      
Feb. 19 Space curves (Manning) Ch 4    
Feb. 21 Surfaces in R^3 (Manning) Ch 5 HW 3 3/7
Feb. 26 Feb. Break (No class)      
Feb. 28 Problem discussion      
Mar. 5 Linear and Quadratic maps      
Mar. 7 Prelim I   HW 4  
Mar. 12 Riemannian metric      
Mar. 14 Tangent space, examples Ch 6, 7    
Mar. 19 Conformal maps, Spherical geometry      
Mar. 21 Hyperbolic geometry      
Mar. 26 Measure, area      
Mar. 28 Volume, isometry   HW 5  
Apr. 2 Spring Break (No class)       
Apr. 4 Spring Break (No class)       
Apr. 9 Examples      
Apr. 11 Vector fields      
Apr. 16 Prelim 2      
Apr. 18 Derivative of functions      
Apr. 23  Covariant derivative      
Apr. 25 Christoffel symbols,  Parallel translation, Curvatures Ch 8    
Apr. 30  Gauss equation,      
May 2 Geodesic equation Ch 9, 10    
May 7 Gauss's Theorema Egregium      
May 16 Final Due