# 401-3531-00L Differential Geometry I

Semester | Autumn Semester 2016 |

Lecturers | U. Lang |

Periodicity | yearly recurring course |

Language of instruction | German |

Comment | This course counts as a core course in the Bachelor's degree programme in Mathematics. Holders of an ETH Zurich Bachelor's degree in Mathematics who didn't use credits from neither 401-3531-00L Differential Geometry I nor 401-3532-00L Differential Geometry II for their Bachelor's degree still can have recognised this course for the Master's degree. Furthermore, at most one of the three course units 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. |

### Courses

Number | Title | Hours | Lecturers | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

401-3531-00 V | Differentialgeometrie I | 4 hrs |
| U. Lang | ||||||||||||

401-3531-00 U | Differentialgeometrie I Do 13-14 oder Do 14-15 oder Fr 13-14 | 1 hrs |
| U. Lang |

### Catalogue data

Abstract | Curves in R^n, inner geometry of hypersurfaces in R^n, curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, tangent bundle, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. |

Objective | Introduction to elementary differential geometry and differential topology. |

Content | - Differential geometry in R^n: theory of curves, submanifolds and immersions, inner geometry of hypersurfaces, Gauss map and curvature, Theorema Egregium, special classes of surfaces, Theorem of Gauss-Bonnet, Poincaré Index Theorem. - The hyperbolic space. - Differential topology: differentiable manifolds, tangent bundle, immersions and embeddings in R^n, Sard's Theorem, transversality, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. |

Literature | Differential Geometry in R^n: - Manfredo P. do Carmo: Differential geometry of curves and surfaces - Wolfgang Kühnel: Differentialgeometrie. Curves-surfaces-manifolds - Christian Bär: Elementary differential geometry Differential Topology: - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds - Victor Guillemin & Alan Pollack: Differential Topology - Morris W. Hirsch: Differential Topology |

### Performance assessment

Performance assessment information (valid until the course unit is held again) | |

Performance assessment as a semester course | |

ECTS credits | 10 credits |

Examiners | U. Lang |

Type | session examination |

Language of examination | German |

Repetition | The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit. |

Mode of examination | oral 30 minutes |

Additional information on mode of examination | Prüfungssprache: Deutsch oder Englisch / Language of examination: English or German |

This information can be updated until the beginning of the semester; information on the examination timetable is binding. |

### Learning materials

Teaching document repository | Lehr-Dokumentenablage / Teaching document repository |

Only public learning materials are listed. |

### Groups

No information on groups available. |

### Restrictions

There are no additional restrictions for the registration. |