2007 April 20 / E-Mail Me

This schedule is under construction; it will be updated as we go along. Unless otherwise noted, the readings and exercises are from *Differential Geometry of Curves and Surfaces* by Manfredo P. do Carmo.

Lecture | Day | Topic | Reading | Problems | Due Date |
---|---|---|---|---|---|

1 | W | introduction | |||

2 | F | parametrized curves, dot product | 1.2 | #1, 2, 5 | 19 Jan |

3 | W | norm, regular curves, arc length | 1.3 | #4, 6, 10 | 26 Jan |

4 | F | cross product, curvature of curves | 1.4 | #10, 11, 12 | 26 Jan |

5 | M | Frenet frame | 1.5 | #2, 3, 4, 5 | 26 Jan |

6 | W | fundamental theorem of curves | 1.5 | Hmwk3, Lab1 | 2 Feb |

7 | F | isoperimetric inequality, calculus of variations | 1.7 | ||

8 | M | Euler-Lagrange equations | Mechanics | Hmwk4 | 9 Feb |

9 | W | action principle | Mechanics | ||

10 | F | curves as 1-manifolds | 2.2 | ||

11 | M | surfaces | 2.2 | Hmwk5 | 16 Feb |

12 | W | examples | 2.2 | ||

13 | F | inverse/implicit function theorems | 2.2 | ||

14 | M | coordinate changes, functions on surfaces | 2.3 | Exam1 | 20 Feb |

15 | W | Morse theory | Wikipedia | ||

16 | F | differentials | 2.Appendix | ||

17 | M | tangent plane | 2.4 | Hmwk6 | 2 Mar |

18 | W | first fundamental form | 2.5 | ||

19 | F | pulling back inner products, arc length | 2.5 | ||

20 | M | integration, area vs. boundary length | 2.5 | #15 | 9 Mar |

21 | W | Gauss map | 3.2 | #6, 8 | 9 Mar |

22 | F | self-adjoint maps, quadratic forms | 3.Appendix | Lab2 | 9 Mar |

23 | M | second fundamental form | 3.2 | ||

24 | W | Gauss/mean/normal curvatures | 3.3 | #6bc, 13 | 23 Mar |

25 | F | isometries, conformal maps | 4.2 | #7, 15, 18 | 23 Mar |

26 | M | Christoffel symbols | 4.3 | #2, 3, 8, and 4.2 #19 | 30 Mar |

27 | W | Theorema Egregium, vector fields | 4.3 | Lab3 | 30 Mar |

28 | F | covariant derivative | 4.4 | #4 | 30 Mar |

29 | M | geodesics | 4.4 | Exam2 | |

30 | W | parallel transport, holonomy | 4.4 | ||

31 | F | examples of abstract surfaces | Manifolds | ||

32 | M | abstract surfaces, change of coordinates | 5.10 | ||

33 | W | Riemannian surfaces | 5.10 | ||

34 | F | group project work | |||

35 | M | hyperbolic plane, non-Euclidean geometry | 5.10 | ||

36 | W | minimal surfaces, soap films | 3.5 | ||

37 | F | harmonic/holomorphic functions | 3.5 | ||

38 | M | geodesic curvature | 4.4 | ||

39 | W | local Gauss-Bonnet theorem | 4.5 | ||

40 | F | Gauss-Bonnet theorem | |||

41 | M | student talks | |||

42 | W | student talks |