**Introduction to Differential Geometry: MATH
4540**

Spring 2017

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

**E-mail: **cao at math.cornell.edu

**Teaching
Assistant:** Daoji Huang (Email:dh539 at cornell.edu 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: **

**Elementary Differential Geometry (2nd Ed.), Andrew Pressley**

**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.

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 | HW 6 | |||

May 2 | Geodesic equation | |||

May 4 | Gauss's Theorema Egregium | |||

May 9 | Survey | |||

May 16 | Final Due | |||