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)

10/21: FALL BREAK

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.

11/4; FIRST EXAM

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.

11/25:THANKSGIVING

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

curvature.

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.