Math 120B: General Course Outline
Catalog Description
    120B. Differential Geometry. (4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 131A. Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.
Textbook
    R. Millman and G. Parker, Elements of Differential Geometry, Prentice-Hall Inc.
Schedule of Lectures

Lecture
Topics
1
Existence and uniqueness theorem for ODE's. Existence of geodesics.
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.
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.
8
Statement of proof of the Gauss-Bonnet theorem, local version.
9
Global surfaces (two-dimensional manifolds).
10
The Euler-Poincare characteristic.
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.

Comments

Outline update: M. Green, 5/96
(Requisites updated 5/98)
For more information, please contact Student Services, ugrad@math.ucla.edu.
 

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