MATH3021

**for Part 1 see the page
of Norbert
Peyerimhoff**

**Lecturer:** Olaf Post

**Term: ** Epiphany 2012/13

- Wednesdays, 9:00 in CG91
- Thursdays, 9:00 in CG93
- starts on Wednesday, 16 January
- ends on Thursday, 14 March

**Literature**

- LM Woodward and J Bolton,
*Differential Geometry Lecture Notes.*Copies are available from the Maths office at 50p per chapter - M doCarmo,
*Differential Geometry of Curves and Surfaces* - Bruce and Giblin,
*Curves and singularities*(looks at curves from a different viewpoint)

**Problem session**

**Assignments**

Homework | Date | Hand in |

Problem Sheet 1 | 07-02-2013 | Questions 60 and 67 on 21-02-2013 |

Problem Sheet 2 | 06-03-2013 | Questions 85 and 86 on 14-03-2013 |

**Content of Lectures**

Date | Content |

Wednesday, 16 January 2013 (Week 11) | Isometries and local isometries, examples of isometries: cylinder and plane |

Thursday, 17 January 2013 (Week 11) | Conformal maps, conformal group and isometry group of a surface |

Wednesday, 23 January 2013 (Week 12) | The Geometry of the Gauss map, the Weingarten map and its symmetry, examples: sphere, surface of revolution |

Thursday, 24 January 2013 (Week 12) | Gaussian curvature, mean curvature, principal curvatures and principal directions, umbilic points, example of surface of revolution |

Wednesday, 30 January 2013 (Week 13) | Calculation of the curvatures in a local parametrisation (linear algebra revision sheet), second fundamental form and its coefficients, example: hyperbolic paraboloid |

Thursday, 31 January 2013 (Week 13) | elliptic, hyperbolic and parabolic points, planar points, some global theorems |

Wednesday, 6 February 2013 (Week 14) | Curves in a surface, normal and geodesic curvature of a curve in a surface, calculation using the second fundamental form and a local parametrisation, asymptotic curve of a surface |

Thursday, 7 February 2013 (Week 14) | Problem session (isometries, normal curvature of a curve in a surface) |

Wednesday, 13 February 2013 (Week 15) | Families of curves on a surface, lines of curvature |

Thursday, 14 February 2013 (Week 15) | Examples of lines of curvature: surface of revolution, orthogonal and principal coordinates |

Wednesday, 20 February 2013 (Week 16) | Theorema Egregium of Gauss, Christophel symbols |

Thursday, 21 February 2013 (Week 16) | Christophel symbols depend only on first fundamental form, proof of the Theorema Egregium of Gauss |

Wednesday, 27 February 2013 (Week 17) | Applications of the Theorema Egregium of Gauss, the Gaussian curvature of the hyperbolic plane |

Thursday, 28 February 2013 (Week 17) | Geodesics and geodesic curvature, calculation in local coordinates |

Wednesday, 6 March 2013 (Week 18) | Local existence and uniqueness of geodesics, invariance under local isometries; examples of geodesics: surface of revolution |

Thursday, 7 March 2013 (Week 18) | Motivation for the Gauss-Bonnet theorem, Euler characteristic |

Wednesday, 13 March 2013 (Week 19) | The Gauss-Bonnet theorem |

Thursday, 14 March 2013 (Week 18) | Problem session |

Last modified: 2013-05-09