# 401-3531-00L Differential Geometry I

Semester | Autumn Semester 2017 |

Lecturers | D. A. Salamon |

Periodicity | yearly recurring course |

Language of instruction | English |

Comment | At most one of the three course units (Bachelor Core Courses) 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. |

Abstract | Submanifolds of R^n, tangent bundle, embeddings and immersions, vector fields, Lie bracket, Frobenius' Theorem. Geodesics, exponential map, completeness, Hopf-Rinow. Levi-Civita connection, parallel transport, motions without twisting, sliding, and wobbling. Isometries, Riemann curvature, Theorema Egregium. Cartan-Ambrose-Hicks, symmetric spaces, constant curvature, Hadamard's theorem. |

Objective | Introduction to Differential Geometry. Submanifolds of Euclidean space, tangent bundle, embeddings and immersions, vector fields and flows, Lie bracket, foliations, the Theorem of Frobenius. Geodesics, exponential map, injectivity radius, completeness Hopf-Rinow Theorem, existence of minimal geodesics. Levi-Civita connection, parallel transport, Frame bundle, motions without twisting, sliding, and wobbling. Isometries, the Riemann curvature tensor, Theorema Egregium. Cartan-Ambrose-Hicks, symmetric spaces, constant curvature, nonpositive sectional curvature, Hadamard's theorem. |

Literature | Joel Robbin and Dietmar Salamon "Introduction to Differential Geometry", https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf |