## OVERVIEW

Below is a list of core topics in the approximate order we will be covering them. This material is roughly the contents of do Carmo, chapters 2-4. Other topics may be treated at the end of the course, if time permits.

- Computations in coordinate charts: first fundamental form, Christoffel symbols.
- Geodesics.
- Submanifolds of Euclidean space. Changes of co-ordinates. Isometries.
- Orthogonal co-ordinates, geodesic polar co-ordinates.
- Gauss map, second fundamental form.
- Theorema egregium. Minding's theorem. Gauss-Bonnet theorem.

## TOPICS BY LECTURE (with do Carmo references)

Wed 7 Sep: |
2.2, 2.3 | Regular parametrized submanifolds. Surfaces of revolution. |

Fri 9 Sep: |
2.5; Munkres | First fundamental form. Volume. |

Mon 12 Sep: |
2.5 | Length. Angles. |

Wed 14 Sep: |
4.4 | Covariant derivative. Parallel vector fields. |

Fri 16 Sep: |
4.4 | Christoffel symbols. |

Mon 19 Sep: |
4.4 | Existence & uniqueness of parallel vector fields. Parallel transport. |

Wed 21 Sep: |
4.4 | Geodesics. Local existence & uniqueness. Sphere, cylinder. |

Mon 26 Sep: |
4.4 | Clairaut's relation. Geodesics of sphere, ellipsoid of revolution, paraboloid, cusp. |

Wed 28 Sep: |
4.4 | Integrating geodesics using Clairaut's relation. Cylinder, cone, paraboloid. |

Fri 30 Sep: |
4.4 | Geodesics of torus of revolution |

Mon 3 Oct: |
Klingenberg | Geodesics of triaxial ellipsoid. |

Wed 5 Oct: |
5.10 | Abstract pseudo-Riemannian manifold patches. Geodesics of hyperbolic space. |

Fri 7 Oct: |
Morgan | Geodesics of Schwarzschild space. Precession of Mercury. |

Wed 12 Oct: |
4.2 | (Local) isometries I: plane, cylinder, Clifford torus in R^4; helicoid, catenoid. |

Fri 14 Oct: |
5.4 | First variation formula. Locally length-minimizing implies geodesic. |

Mon 17 Oct: |
2.2, 2.3 | Submanifolds. |

Wed 19 Oct: |
2.2, 2.3 | Regular level sets. Changes of co-ordinates. |

Mon 24 Oct: |
2.4, 4.4 | Tangent space. Covariant derivative, geodesics on submanifolds. |

Wed 26 Oct: |
2.5 | Change of co-ordinates formulas for first fundamental form and Christoffel symbols. |

Fri 28 Oct: |
4.2 | (Local) isometries II: plane, cone; hyperbolic plane, tractroid |

Mon 31 Oct: |
1.3, 3.4 | Arc-length parametrization of curves; orthogonal parametrizations of surfaces. |

Wed 2 Nov: |
3.4, 3.5, 4.2 | Orthogonal and isothermal parametrizations of surfaces. |

Fri 4 Nov: |
4.6, 4.7 | Exponential map, normal and geodesic polar co-ordinates. |

Mon 7 Nov: |
4.6 | Gauss lemma. Geodesics are locally distance-minimizing. |

Wed 9 Nov: |
2.6 | Equal-area parametrizations. Orientations. |

Mon 14 Nov: |
2.6, 3.2 | Orientations. Gauss map. |

Wed 16 Nov: |
3.2 | Differential of Gauss map. Principal, Gauss, mean curvatures. |

Fri 18 Nov: |
1.5, 3.2, 3.3 | Curvature of some examples. Plane curves. Surfaces of revolution. |

Mon 21 Nov: |
3.2, 3.3 | Proposition 2.3.2. Second fundamental form. Differential of Gauss map is self-adjoint. Normal sections. |

Wed 23 Nov: |
3.2, 3.3 | Elliptic, parabolic, hyperbolic, planar points. |

Mon 28 Nov: |
4.3 | Theorema egregium. Gauss curvature of surfaces in R^N. |

Wed 30 Nov: |
4.6 | Curvature in geodesic polar co-ordinates. Minding's theorem. |

Fri 2 Dec: |
4.5, 5.7 | Brouwer degree. Hopf rotation angle theorem. Gauss-Bonnet theorem. |

Mon 5 Dec: |
4.4, 4.5 | Covariant derivatives of unit vector fields, geodesic curvature. (Local) Gauss-Bonnet formula. |

Wed 7 Dec: |
4.5 | Euler characteristic. (Global) Gauss-Bonnet formula. Applications. |

Fri 9 Dec: |
5.4 | Second variation formula. Bonnet's theorem. |

Mon 12 Dec: |
3.5 | Ruled surfaces, developable surfaces. |