## Math 136: Differential Geometry (Fall 2021)

**Instructor:** Dori Bejleri (bejleri [at] math [dot] university [dot] edu)

**Time and place:** Wednesdays and Fridays at 12:00pm - 1:15pm in Science Center 507

**Syllabus**

**Canvas**

### Lectures

Date | Topic | Text | Notes |
---|---|---|---|

09/03/2021 | Introduction, parametrized curves | [Kü] 2A | Lecture 1 |

09/08/2021 | Plane curves, space curves, Frenet curves | [Kü] 2B-2C | Lecture 2 |

09/10/2021 | Frenet space curves and spherical curves | [Kü] 2C | Lecture 3 |

09/15/2021 | Frenet equations, fundamental existence and uniqueness theorem | [Kü] 2D | Lecture 4 |

09/17/2021 | Fundamental theorem continued, global theory of plane curves | [Kü] 2F | Lecture 5 |

09/22/2021 | Regular surfaces | [Kü] 3A, [DC] 2-2 | Lecture 6 |

09/24/2021 | Regular surfaces continued, first fundamental form | [Kü] 3A, [DC] 2-2, 2-5 | Lecture 7 |

09/29/2021 | First fundamental form, length, area | [Kü] 3A, [DC] 2-5 | Lecture 8 |

10/01/2021 | Area continued, vector fields, orientation | [Kü] 3A, [DC] 2-5 | Lecture 9 |

10/06/2021 | The Gauss map and second fundamental form | [Kü] 3B, [DC] 3-2 | Lecture 10 |

10/08/2021 | Curvature of surfaces | [Kü] 3B, [DC] 3-2, 3-3 | Lecture 11 |

10/13/2021 | Curvature continued | [Kü] 3B, [DC] 3-2, 3-3 | Lecture 12 |

10/15/2021 | Surfaces of revolution, ruled surfaces | [Kü] 3C, [DC] 3-3 | Lecture 13 |

10/20/2021 | Directional and covariant derivatives | [Kü] 4A | Lecture 14 |

10/22/2021 | Christoffel symbols, isometries | [Kü] 4A, [DC] 4-2 | Lecture 15 |

10/27/2021 | Theorem Egregium, Gauss and Codazzi-Mainardi equations | [Kü] 4C, [DC] 4-3 | Lecture 16 |

10/29/2021 | Gauss and Codazzi-Mainardi equations continued, parallel transport | [Kü] 4B | Lecture 17 |

11/03/2021 | Geodesics | [Kü] 4B, [DC] 4-4 | Lecture 18 |

11/05/2021 | Geodesics continued | [Kü] 4B, [DC] 4-4 | Lecture 19 |

11/10/2021 | Riemannian curvature | [Kü] 4C | Lecture 20 |

11/12/2021 | Riemannian curvature continued | [Kü] 4C | Lecture 21 |

11/17/2021 | The fundamental theorem of local surfaces, the exponential map | [Kü] 4D | Lecture 22 |

11/19/2021 | The Gauss-Bonnet theorem | [Kü] 4F | Lecture 23 |

12/01/2021 | The Gauss-Bonnet theorem continued | [Kü] 4F | Lecture 24 |

**Midterm project.**Due October 29.

**Problem Sets**

1. Problem set 1. Due September 17.

2. Problem set 2. Due September 24.

3. Problem set 3. Due October 1.

4. Problem set 4. Due October 8.

5. Problem set 5. Due October 15.

6. Problem set 6. Due October 22.

7. Problem set 7. Due November 5.

8. Problem set 8. Due November 12.

9. Problem set 9. Due November 22.

**Textbook**

[Kü] Kühnel,

*Differential Geometry: Curves-Surfaces-Manifolds.*Third Edition, 2015

[DC] Do Carmo,

*Differential Geometry of Curves and Surfaces.*Second Edition, 2016