Fall 2015 MATH 240 001 LEC

Riemannian Geometry
Units/CreditFinal Exam GroupEnrollment
412: WEDNESDAY, DECEMBER 16, 2015 7-10PLimit:24 Enrolled:11 Waitlist:0 Avail Seats:13 [on 10/04/15]
Additional Information: 

Prerequisites: 214.

Syllabus: Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes.

Office: 705 Evans

Office Hours: by appointment

Literature: M. P. do Carmo, Riemannian geometry (required)
P. Petersen, Riemannian geometry (required), available in hardcopy or online through SpringerLink
S. Sakai, Riemannian geometry (recommended)
J. Cheeger, D. Ebin, Comparison theorems in Riemannian geometry (recommended)

In the first half of the class we will mainly use do Carmo's book, although Petersen's book can be read as well. Later we will talk about a selection of topics that can be found in Petersen's book or in Cheeger and Ebin's book. Each week, I will announce which chapters of which book we are going to talk about. It is recommended that the students read this chapters to review the material from class.

Grading: The grade will be determined by the overall score on the problem sets (60%) and by a 20-30min talk that will be given by each student during reading week (40%).

Problem Sets: Problem sets will be published weekly and are due in class by the specified date. Only 1-2 problems per problem set will be graded. I will often just refer to certain problems in do Carmo or Petersen. So it is helpful to have access to these textbooks. Problem sets are available on https://www.dropbox.com/sh/h6vp3xx97x5vxpr/AAD4mu2SGgLfiT9kXlFMTvuma?dl=0 

Course Outline (subject to change)

  1. Introduction, metric spaces, length spaces, manifolds
  2. Introduction, metric spaces, length spaces, manifolds
  3. vector bundles, tangent space, vector fields, 
  4. tangent space
  5. vector fields, vector bundles continued
  7. differential forms, exterior derivative, Lie bracket, Lie derivative
  8. Riemannian metrics (do Carmo, chapter 1)
  9. connections on vector bundles, Levi-Civita connections (do Carmo, chapter 2)
  10. geodesics, convex neighborhoods (do Carmo, chapter 3)
  11. curvature (do Carmo, chapter 4)
  12. curvature
  13. Jacobi fields (do Carmo, chapter 5)
  14. isometric immersions (do Carmo, chapter 6)


Course Webpage: bcourses