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. |