MS 6617 E
Office
hours: Monday: 12:30pm-12:55pm , 2pm-2:30pm----- Wednesday : 12:00-12:55 pm , 2:00-3:00 pm
120B. Differential Geometry. (4) Lecture, three hours; discussion, one hour. Topics in surface theory and higher dimensional differential geometry .
Class meeting time: Monday, Wednesday and Friday , 1-1:50 pm in MS 5148
Textbook: R. Millman and G. Parker, Elements of Differential Geometry, Prentice-Hall Inc.
Paper | 20% |
Homework | 40% |
Take home Final Exam | 40% |
Check here often for assignments and other information.
Lecture |
Topics |
1 |
Existence and
uniqueness theorem for ODE's. Existence of geodesics. Variation of Arc Length and Geodesics--click here , then print Here is HW #1 , click here, then print. Due on Monday April 21st. |
2 |
Geodesic polar
coordinates. Orthogonality of geodesic polar coordinates. Formula for
Gaussian curvature in geodesic polar coordinates. |
3 |
Geodesics are
locally shortest paths. Estimate on lengths and areas of geodesics
circles in terms of curvature. Geodesic Normal Coordinates on a Surface-- class notes |
4 |
Local
isometry. Surfaces of constant curvature are locally isometric to
standard spaces. |
5 |
Surfaces of
revolution of constant curvature. |
6 |
Classification
of surfaces with Gaussian curvature equal to zero (cylinders, cones,
developable surfaces). |
7 |
Formula for
geodesic curvature in geodesic polar coordinates. Gauss Curvature and the Form of the Metric in Geodesic Polar Coordinates Why T(N(r))=0 for a surface in R3--notes for the end of the April 21 lecture. More on the Form of the Metric in Geodesic Normal Coordinates Here is HW #2 |
8 |
Statement of
proof of the Gauss-Bonnet theorem, local version. |
9 |
Global
surfaces (two-dimensional manifolds).
Here is HW #3 . Topics for Paper The Gauss Map and the Gauss Curvature . |
10 |
|
11 |
The global
Gauss-Bonnet theorem. Degree of the Gauss map. |
12 |
The global
Gauss-Bonnet theorem: examples. |
13 |
Vector fields.
Local index of a vector field. |
14 |
Total index of
a vector field. Poincare-Hopf theorem. |
15 |
Hilbert's
rigidity theorem for the sphere. |
16 |
Introduction
to manifolds. Examples. |
17 |
Tangent space
to a manifold at a point. |
18 |
Differentiable
maps between manifolds. Derivative of a map between
manifolds. |
19 |
Vector fields
and Lie brackets. |
20 |
Riemannian
metrics. Geodesics revisited. |
21 |
Connections.
The connection associated to a Riemannian manifold. |
22 |
The hyperbolic
plane (=Poincare disk). |
23 |
Parallel
transport. Geodesics using connections. |
24 |
Curvature of a
Riemannian
manifold. |