Math 567 differential geometry- topics covered

8/26: definition of regular surface. Ex: spherical and stereographic
coords. on the sphere. Every surface is locally a graph.

8/31 Ex: torus of revolution. Preimages of regular values. (Ex: torus).
Surfaces of revolution.

9/2 Calculus on surfaces: coordinate changes on sfcs. Differentiable
functions and maps. Tangent plane, differential of a function.

9/7 Def. of differentiable manifold; diff'ble functions and maps between manifolds.
Vector fields (3 equivalent definitions).

9/9 1-forms in R^n and on manifolds. Riemannian metric on manifolds; first
fundamental form of surfaces (=induced metric). Ex; helicoid

9/14 lengths of curves on surfaces and Riemannian manifolds; area of
regions on surfaces. 2 equivalent definitions of orientability.

9/16 The Mobius band is non-orientable. Second fundamental form of surfaces;
normal curvature of curves on surfaces

9/21 Numerical invariants: principal curvatures, mean curvature, Gauss curvature.
Ex: hyperboloid (saddle). Principal directions, umbilic points. Surfaces
with all points umbilic.

9/23 Elliptic, hyperbolic, parabolic and planar points. The second f.f. in local
coordinates. Examples: torus, surfaces of revolution, graphs.

9/28 Geometric interpretation of 2nd. f.f: Dupin indicatrix, Hessian.
Mean curvature of a graph in "divergence form". Example: catenoid.
"Third fundamental form"

9/30 Problem session: orientability "pulls back but does not push forward"
under a map; differentiable covering maps (definition, examples).
Average of a quadratic form on the unit sphere.

10/5 Problem session: umbilic points of an ellipsoid; surfaces of rev'n
with constant curvature. Gauss's interpreation of curvature. The
Gauss map of a minimal surface is conformal.

Isometries between surfaces; local isometries.

10/7 Examples of local isometries: cylinder and plane, cone and plane,
helicoid and catenoid. Directional derivatives of vector fields on surfaces

10/12 covariant derivative on surfaces; computation of Christoffel symbols from the
first f.f.; the second f.f. as "normal component" of the directional derivative.

10/14 Necessary conditions to realize the 1st. and 2nd. ff's by an immersion;
The Gauss and Mainardi-Codazzi  equations. Gauss's "Teorema Egregium".

10/19 Integrability conditions; solvability of 1st. order systems of PDE.
Proof of the main local theorem of surface theory (Bonnet)


10/26 Problem: when are two vector fields the tangent vector fields of a
diffeomorphism? Def. of Lie bracket of vector fields. Proof of a
special case of Frobenius' theorem.

10/28: Def. of connection; torsion-free and metric-compatible connections.
Levi-Civita's theorem. (statement).  Existence of parallel o.n. frames implies
local isometry to euclidean. Tensors on manifolds   Def. of the Riemann curvature tensor.

11/2: Invariance of Lie bracket under diffeomorphisms.  Coordinate-free statements of the
Gauss and Mainardi-Codazzi equations. Thm.: surfaces with vansihing Gauss
curvature are locally isometric to the Euclidean plane.


11/9: minimal surfaces: first and second variations of area. The Laplace- Beltrami
operator and the metric gradient.

11/11 minimal surfaces:  position vector is g-harmonic; conjugate harmonic functions
existence of  isothermal coordinates on minimal surfaces .

11/16  minimal surface: position vector in isothermal coordinates is harmonic.
The Cauchy-Riemann equaions. Minimal surfaces and  analytic maps to C^3.

11/18 minimal surfaces and analytic functions: the Weierstrass representation formula
The Gauss map as a meromorphic function.

11/23: minimal surfaces: the minimal surface equation, the maximum principle for
minimal surfaces, some applications.


11/30 : Gauss-Bonnet theorem via orthonormal frames: o.n frames and connection
one-forms,  Gauss curvature and exterior derivative of connection one-form, geodesic

12/2 change of frames and the angle formula.
Winding number of a curve in the plane. Hopf's "turning tangents" theorem. Local
Gauss-Bonnet theorem for smooth domains.

12/7  Extensions of Hopf's theorem to general metrics and pw C^1 curves.
Local Gauss-Bonnet for regions bounded by pw C^1 curves.  "Angle excess"
for geodesic triangles. Index of vector fields
with isolated singularities. Global Gauss-Bonnet for surfaces without boundary.

12/9 Euler characteristic of compact surfaces: equivalent def's via vector fields and
triangulations. Euler characteristic =2-2(genus).  Gauss-Bonnet for surfaces with
boundary.  Def. of geodesics and parallel transport. Examples: geodesics on the
sphere and parallel transport around a parallel on the sphere.  Parallel transport
around a simple loop and  integral of  Gauss curvature.