Introduction to Differential Geometry: MATH 4540
Spring 2017

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

E-mail: cao at

Teaching Assistant: Daoji Huang (Email:dh539 at OH: Tuesday & Thursday 3:00-4:00pm, 218 MLT)

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

Office Hour:Tuesday & 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 Thursday, March 9th and Thursday, April 13th.   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 Tuesday, 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 (Temtative!)

Class Topic Read   Exercises Due
Jan. 26 Introduction, definition of maps, curves Ch 1 HW 1 2/9
Jan. 31 Examples, arc length, curvature Ch 2
Feb. 2 Derivative of maps Ch 3
Feb. 7 Composition of maps HW 2 2/23
Feb. 9 Space curves, curvature
Feb. 14 Proper maps, inverse function theorem
Feb. 16 Properties and examples HW 3 3/9
Feb. 23 Inverse function theorem, implicit function theorem,
Feb. 28 Examples of surfaces Ch 4
Mar. 2 Surfaces, Isoperimetric Inequality (I) HW 4
Mar. 7 Linear and Quadratic maps
Mar. 9 Prelim I
Mar. 14 Riemannian metric
Mar. 16 First fundamental form, isometry HW
Mar. 21 Conformal maps
Mar. 23 Spherical geometry Ch 6, 7
Mar. 28 Hyperbolic geometry
Apr. 5 Measure, area
Apr. 7 Introduction to curvature, Christoffel symbols Ch 8
Apr. 13 Prelim 2 HW 5
Apr. 18 Derivative of functions
Apr. 20 Covariant derivative
Apr. 25 Gauss equation, Parallel translation Ch 9, 10
Apr. 27 Curvatures HW 6
May 2 Geodesic equation
May 4 Gauss's Theorema Egregium
May 9 Survey
May 16 Final Due