Spring 2011

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**(5/16)**The final version of the notes on the intrinsic proof of the Gauß-Bonnet theorem is here.**(5/10)**I just uploaded a corrected version of the notes on the intrinsic proof of the Gauß-Bonnet theorem.**(5/6)**I just posted my notes on the intrinsic proof of the Gauß-Bonnet theorem.**(5/6)**Here are some hints for homework 5.**(4/22)**Homework 5 has been posted. It's due on the last day of class, May 16.**(4/21)**I just posted a handout on the following question: suppose X_{1},..., X_{n}are smooth linearly independent vector fields on a smooth n-manifold M. When are they locally coordianate vector fields? Answer: when they mutually commute. This will be used on Monday to finish the proof that R = 0 iff M is flat.**(3/22)**Homework 4 has been posted. It's due approximately on April 13.**(3/7)**The class is moving from CL 127 to**Sweeney Hall 240**. We'll first meet there this Wednesday.**(3/5)**I posted some hints for homework 3.**(3/1)**Homework 3 has been posted and is due somehwhere around March 16.**(2/28)**As promised, I posted a link to a short and elementary introduction to differential forms.**(2/15, part 2)**I just posted a list of some potential topics for the survey paper. More to come. Also, feel free to suggest others.**(2/15)**I have posted a short write-up called Three faces of the Lie bracket. It describes three equivalent characterizations of the Lie bracket of vector fields.**(2/11, part 2)**I just posted a link to a handout on the coordinate-free view of the derivative of a map between Euclidean spaces.**(2/11)**I just posted Homework 2 roughly due on Febraury 21.**(2/2)**I left copies of relevant chapters from Boothby in an envelope hanging near my office door - help yourselves.**(2/1)**My office hours have changed. They are now MW 11:45-12:45 + 3-4, Tu 2:30-3:30 (online), and by appointment.**(1/25)**Welcome to the Math 213B home page!

- Riemannian geometry is a generalization of the classical differential geometry of curves and surfaces you studied in Math 113 (or an equivalent course) to abstract smooth manifolds equipped with a family of smoothly varying inner products on tangent spaces. The main goal of Riemannian geometry is to understand how a manifold curves in different directions and how curvature of a manifold relates to its topology.

- Math 213A or
**instructor consent**. **Note**: To make the course more accessible, I will not assume familiarity with abstract smooth manifolds. Instead, I will give a brief introduction to basics topics in smooth manifolds, which are necessary for this course. I'm willing to accept anyone with a basic background in geometry, multivariable calculus, and linear algebra, and a certain level of mathematical maturity.

Tu 2:30-3:30 (online), and by appointment

John M. Lee: Riemannian Manifolds, An Introduction to
Curvature, Springer, GTM vol. 176, 1997 |

- Green sheet

Introduction, review and motivation. Basics of smooth
manifolds, vector bundles and tensors. Definitions and examples of
Riemannian metrics. Connections. Riemannian geodesics. Geodesics and
distance. Curvature. Riemannian submanifolds. The Gauss-Bonnet
theorem. Jacobi fields. Curvature and topology.

# | Date | Sections to be covered |
---|---|---|

1 | 1/26 | Ch. 1, Intro. to curvature |

2 | 1/31 | Smooth manifolds (Boothby) |

3 | 2/2 | Smooth maps (Boothby) |

4 | 2/7 | Immersions and embeddings (Boothby) |

5 | 2/9 | Tangent bundle and vector fields (Boothby) |

6 | 2/14 | Vector bundles. Tensors (Boothby) |

7 | 2/16 | Differential forms (Boothby) |

8 | 2/21 | Ch. 3: Riemannian metrics and elementary constructions |

9 | 2/23 | Ch. 3: Generalizations of Riemannian metrics and model spaces |

10 | 2/28 | Ch. 4: Motivation, connections |

11 | 3/2 | Ch. 4: Vector fields along curves |

12 | 3/7 | Ch. 4: Geodesics |

13 | 3/9 | Ch. 5: The Riemannian connection. Exponential map. |

14 | 3/14 | Ch. 5: Normal coordinates. Geodesics of model spaces |

15 | 3/16 | Ch. 6: Lenghts and distances on Riem. manifolds. Geodesics as minimizers |

16 | 3/21 | Ch. 6: Completeness and Hopf-Rinow |

17 | 3/23 | Ch. 7: Local invariants and flat manifolds |

* | 3/28-4/1 | Spring break |

18 | 4/4 | Ch. 7: Symmetries of the curvature tensor |

19 | 4/6 | Ch. 7: Ricci and scalar curvatures |

20 | 4/11 | Ch. 8: Riem. submanifolds and the second fundamental form |

21 | 4/13 | Ch. 8: Hypersurfaces in Euclidean space |

22 | 4/18 | Ch. 8: Curvature in higher dimensions |

23 | 4/20 | Ch. 9: Gauss-Bonnet formula |

24 | 4/25 | Ch. 9: Gauss-Bonnet theorem |

25 | 4/27 | Ch. 10: Jacobi equation and computation of Jacobi fields |

26 | 5/2 | Ch. 10: Conjugate points and second variation formula |

27 | 5/4 | Ch. 10: Geodesics do not minimize past conjugate points |

28 | 5/9 | Ch. 11: Some comparison theorems. Manifolds of negative curvature |

29 | 5/11 | Ch. 11: Manifolds of positive curvature |

30 | 5/16 | Ch. 11: Manifolds of constant curvature, the end! |

- Homework 40%, expository paper 30%, take-home final 30%.

- Basic topology
- A coordinate-free view of the derivative
- Three faces of the Lie bracket
- What is a differential form?
- When are vector fields coordinate vector fields?
- An intrinsic proof of the Gauß-Bonnet theorem

- There will be regular homework assignments, taken mostly from the textbook.

# | Due date | Assignment |
---|---|---|

1 | 2/7 | HW 1 |

2 | 2/21 | HW 2 |

3 | 3/16 | HW 3 |

4 | 4/13 | HW 4 |

5 | 5/16 | HW 5 |

- Each student is expected to write a short expository paper in
LaTeX on a topic related to Riemannian geometry. Here are some
potential topics:
- Subriemannian geometry
- Semi- (or pseudo-) riemannian geometry and relativity
- Geodesic flow of a surface of negative curvature
- The Frobenius theorem on integrability of distributions
- Ricci flow: definition and basic properties
- Curvature and fundamental group
- Foliations
- Principal bundles
- Integration of differential forms on manifolds
- Etc.

Slobodan N. Simić

Last modified: Mon May 16 15:25:24 PDT 2011