18.950 - Differential Geometry (Fall 2016)


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.

Course syllabus and policies

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.