Math 439. Introduction To Differential Geometry - Fall 18 - Hans Lindblad

Syllabus Differential Geometry can be seen as continuation 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. The theory of the curvatures was used in Einstein's theory of General Relativity to study how space-time curves under the influence of gravity.
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
Exams/Grades: Exam 1 (15%), Exam 2 (15%), Exam 3 (15%), Final 12/12 at 9-12 am (30%), Homework (25%).
Professor: Hans Lindblad, lindblad@math.jhu.edu, office hour Th 2-3 in Krieger 406
TA: Dan Ginsberg, dginsbe5@math.jhu.edu, office hour M 5-6 in Krieger 202.
Preliminary lecture schedule: TuTh 3.00-4.15 in Maryland 104.
wk  date  Tuesday  Thursday
  1  9/3  1.1 Introduction,1.2-3 Parametrizedcurve-arclength  1.4 Vectorproduct,1.5 Characterization of curves
  2  9/10  2.2, App B Def. Regular Surface, Differential  2.2 Regular Surfaces
  3  9/17  2.3 Differentiable map on a surface  2.4 Tangent plane and differential
  4  9/24  Contractions, Inverse function theorem  Exam 1
  5  10/1  2.5-6 First Fundamental form, Orientation  3.2 Second Fundamental form
  6  10/8   3.2, Appendix A  3.3 The Gauss map
  7  10/15   Exam 2
  8  10/22  4.2 4.3
  9  10/29 4.4 4.4,4.5
10  11/5   Appendix, 4.6   Exam 3
11  11/12  5.3  5.4
12  11/19  No class  No Class
13  11/26 5.2 5.10
14  12/3 Riemann Geometry Riemann Geometry
Preliminary homework schedule. Homework due Tuesday 10 am in mailbox door of Kriger 213 or by email to TA
wk  date  Homework (preliminary)  Solutions
  1  9/4  Review Vector Calculus, dot and cross product, normals, parametrized curves and arclength.  
  2  9/11   1.2: 4,   1.3: 2,4,10,   1.4: 2,6,12,   1.5: 1,2,5,7,8,12,  hw1.pdf
  3  9/18   1.7: 1,15a,   2.2: 1,3,6,7,8,10,11,13,16,  hw2.pdf
  4  9/25   2.3: 1,2,3,8,   2.4: 1-3,7,9,13a,16,17,24,26,  hw3.pdf
  5  10/2    
  6  10/9  2.5: 1,3,5,9,11  
  7  10/16  3.2: 1,4,5,6,7,8,13,14,16,17,   3.3: 1,2,3,4,6,7  
  8  10/23    
  9  10/30  4.2: 1,3,4,5,6,10,   4.3: 1,3,4,6,7,8 ;  
10  11/6  4.4: 1,2,5,8,9,15a  
11  11/13    
12  11/20  No Homework  
13  11/27   TBA  
14  12/4   TBA