Date |
Topics |

W 1/4 | Manifolds |

F 1/6 | Tangent spaces. Vector fields. Integral curves. |

W 1/11 | Flows. Lie derivative. Lie bracket. Cotangent spaces. |

F 1/13 | 1-forms. Tensorial property. Moving frames. Cartan's formula. |

W 1/18 | Tensors. Differential forms. Affine connection. Torsion. |

F 1/20 | Parallel transport. Tangent bundle. Horizontal subspaces. Geodesics and spray. |

W 1/25 | Geodesic flow. Affine parameters. Exponential map. Curvature. Jacobi equation. |

F 1/27 | More on torsion and curvature. |

W 2/1 | Newtonian spacetime. Minkowski spacetime. Lorentz group. Minkowski metric. Proper time. |

F 2/3 | Lorentzian metric. Causal characters. First variation of proper time. Levi-Civita connection. |

W 2/8 | Geodesic principle. Ricci tensor and scalar. Einstein field equations. Stress-energy tensor. |

F 2/10 | Harmonic coordinates. Scalar fields. Curvature identities. |

W 2/15 | Vector fields along maps. Pull-back connections. |

F 2/17 | Second variation of length. Index form. Jacobi fields. |

W 2/22 | Fundamental lemma. Conjugate points. |

F 2/24 | Jacobi's theorem. |

2/27–3/3 | Reading week |

W 3/8 | Null geodesics. |

F 3/10 | Conjugate points for null geodesics. |

W 3/15 | No class. |

F 3/17 | Timelike geodesic congruences. Raychaudhuri equation. Focusing theorem. |

W 3/22 | Hypersurface orthogonal geodesics. |

F 3/24 | Null geodesic congruences. |

W 3/29 | Road map for singularity theorems. Causality. |

F 3/31 | Semicontinuity of proper time. Local maximality of geodesics. Gauss lemma. |

W 4/5 | Inextendible curves. Global hyperbolicity. |

F 4/7 | Locally Lipschitz curves. Accumulation of causal curves. |

M 4/10 | Course project presentations, 13:00–15:00, Burnside Hall 1028 |

M 4/24 | Course project presentations, 10:00–16:00, Burnside Hall 1120 |

**Instructor:** Dr. Gantumur Tsogtgerel

**Tentative plan:**

**Grading:** Homework 50% and the final project 50%.