Text:
Do Carmo
Differential Geometry of Curves and Surfaces
Other books:
Oprea Differential Geometry and Its Applications,
O'Neill Elementary Differential Geometry.
Thorpe Elementary Differential Geometry,
Spivak A comprehensive introduction to Differential Geometry II-III
Lectures MWF 2-3 HSS 1305.
Instructor Hans Lindblad, lindblad@math.ucsd.edu. Office hour M 3-4 APM 7220.
Sections Tu 1-2 APM B412.
TA Patrick Driscoll pdriscol@math.ucsd.edu. Office hour F 12-1 Tu 11-12 APM 6351.
Syllabus
Differential Geometry can be seen as continuations of Vector Calculus.
Geometry is the part of
mathematics concerned with questions of size, shape and position
of objects in space. Differential geometry uses the methods of
differential calculus to study the geometry.
A curve in the plane is determined
by its initial point and direction and the curvature at each point
along the curve, that measures how fast the curve pulls away from
the tangent line. Similarly a surface is determined by its curvatures
in different directions at each point on the surface. The curvatures
is the main focus of the course, these are essentially the coefficients
in the second order terms in the Taylor expansion of a surface, or the second
order derivatives of the parameterization of a surface.
The theory of the curvatures was later used in Einstein's theory of General
Relativity to study how space-time curves under the influence of gravity.
150B will deal with
1) Further properties of geodesics, 2) Calculus on manifolds with differential forms
and
3) Riemannian Geometry and General Relativity.
| wk | date | Monday | Wednesday | Friday |
| 1 | 1/5 | 1.1 Introduction-overview | 1.2-3 Parametrized curve-arclength | 1.5 Fundamental theorem of curves |
| 2 | 1/12 | 1.7 Isoperimetric inequality | 2.2 Regular Surfaces | App. B Differential, 2.2 Surfaces |
| 3 | 1/19 | Holiday | 2.3 Differentiable map on a surface | 2.4 Tangent plane and differential |
| 4 | 1/26 | 2.4 Differential, Contractions | Inverse Function Theorem | Practice Midterm 1 |
| 5 | 2/2 | Midterm I | 2.5-6 First fundamental form-Orient | 3.2 Second fundamental form |
| 6 | 2/9 | 3.2-App A Second fund. form | 3.2-App A Second fund. form | 3.3 |
| 7 | 2/16 | Holiday | 3.3 | Practice Midterm 2 |
| 8 | 2/23 | Midterm II | 4.2 | 4.3 |
| 9 | 3/2 | 4.3 | 4.4 | 4.4 |
| 10 | 3/9 | 4.4-4.5 | Practice Final | Practice Final |