Riemannian Geometry

Course Description and Grading Breakdown
The plan for the course would be covering basic concepts of Riemannian Geometry such as Riemannian metrics, connections, geodesics, curvature, completeness, exponential map, Jacobi fields and so on.  We plan to prove the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet theorem (expressing the total curvature of a surface in terms of its topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature).

Grading: HW 50%; Final 50%

Course Meeting Time and Location
Monday, Wednesday and Friday
1:00 - 1:55 pm
131 Math Building (Building 15)

Course Instructor Contact Information and Office Hours
125 Math Building (Building 15)
Office Hour: Thursday 10:30 am-11:30 pm, 125 Math Building (Building 15)

TA Contact Information and Office Hours


Course Schedule and Textbook
Textbooks:
1. Riemannian Geometry, Manfredo Do Carmo, ISBN:  978-0817634902.
2. Riemannian Manifolds:  An Introduction to Curvature, John M. Lee,  ISBN-13: 978-0387983226
    (available through the Caltech library here https://www.library.caltech.edu/eds/detail?db=cat04350a&an=caltech.909376.)
I will mainly follow from Do Carmo's book during the first half of the term. In the end of the term, I will use John Lee's book more. 

 DateTopic 
 1/3 W 
 1/5 F 
 1/8 M 
 1/10 W 
 1/12 F
 1/15 M
 Martin Luther King Day (Institute holiday)
 1/17 W 
 1/19 F 
 1/22 M 
 1/24 W 
 1/26 F 
 1/29 M 
 1/31 W 
 2/2  F 
 2/5 M 
 2/8 W 
 2/10 F 
 2/12 M 
 2/14 W 
 2/16 F 
 2/19 M President’s Day (Institute holiday)
 2/21 W 
 2/23 F 
 2/26 M 
 2/28 W 
 3/2 F
 3/5 M           
 3/7 W 
 3/9 F 


Course Policies
The due date for homework is Friday 1pm (at the start of class). Collaboration is allowed for the homework but you have to write your own solution. 

You are allowed to turn in one late homework (without penalty) in the quarter. However, you must inform me about turning in your homework late BEFORE the due date, and you must turn in the late homework within a week of the due date. In general, if the homework is late within a week of the due date, it is still acceptable but will be graded based on 20% points off. 


Assignments
***In case you don't have the textbooks in your hand, I have included the scanned page of the problems appearing in the homework. 
 Date PostedAssignment Due Date 
 Jan 10         HW 1 Jan 19
 Jan 23          HW 2 Feb 2
 Feb 5        HW 3 Feb 16
  Topics for final   
 


Final Exam


Collaboration Table
 HomeworkExams
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
InternetYESNO
Your notes (taken in class)YESYES
Class notes of othersYESNO
Your hand copies of class notes of othersYESYES
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESYES
Your returned homework / examsYESYES
Solutions to homework / exams (posted on webpage)YESYES
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:

CalculatorsYES*NO
ComputersYES*NO

* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.