# Fall 2015 MATH 240 001 LEC

Section | Days/Time | Location | Instructor | CCN |
---|---|---|---|---|

001 LEC | MWF 1-2P | 5 EVANS | BAMLER, R H | 54494 |

Units/Credit | Final Exam Group | Enrollment |
---|---|---|

4 | 12: WEDNESDAY, DECEMBER 16, 2015 7-10P | Limit:24 Enrolled:11 Waitlist:0 Avail Seats:13 [on 10/04/15] |

**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)

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

**Course Webpage:** bcourses