Riemannian Geometry

Course Description and Grading Breakdown

The course will cover basic Riemannian geometry: geometry of Riemannian manifolds, connections, curvature, Bianchi identities, completeness, geodesics, exponential map, Gauss’s lemma, Jacobi fields, Lie groups, principal bundles, and characteristic classes. A main goal will be to prove a number of local-to-global theorems relating curvature to 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 hopefully a special case of the sphere theorem (showing that a manifold with pinched curvature is homeomorphic to a sphere).

Grading: HW 50%; Final 50%

Course Meeting Time and Location
Monday, Wednesday and Friday
1:00 - 1:55 pm
187 Linde

Course Instructor Contact Information and Office Hours
smillie at caltech dot edu
104 Linde Hall
Office Hours: Tuesday 10:30-11:30

Course Schedule and Textbook

Books used in the course:
1. Riemannian Geometry, Manfredo Do Carmo (on reserve at Sherman Fairchild Library)
2. Riemannian Manifolds:  An Introduction to Curvature, John M. Lee http://caltech.tind.io/record/909376?ln=en
3. Riemannian Geometry, Peter Petersen http://caltech.tind.io/record/907769?ln=en

 1/7 Manifolds
 1/9 Tangent bundle
 1/11 Tensors and the metric
 1/14 Partitions of unity, differential forms
 1/16 Lie derivatives and the Frobenius theorem
 1/18 Immersions, submersions, Stokes' theorem
 1/23 Lie groups and frames
 1/25 Connections and geodesics

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. 

 Date PostedAssignment Due Date 
 1/9homework 1 Friday 1/18

Final Exam

Collaboration Table
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
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:


* 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.