Introduction to Differential Geometry, Math 474  
Time and place: Engineering B#, MWF noon-12:50 pm.
Office hours:  MWF 11-12.
 
Main textbook: Manfredo P. Do Carmo, Differential geometry of curves and surfaces, (Prentice-Hall)
Supplementary textbook: Harley Flanders, Differential forms with applications to physical sciences, Dover
Prerequisites: Math 261 and Math 369.
 
Class consists of lectures and weekly homeworks. Homeworks are due on the date specified when the class starts.
Absolutely no credit will be given for late homeworks, no matter what the excuse may be. Please plan accordingly. Early homework delivery is OK (email, fax, drop off at the Department etc.)
However, two homeworks (per student) with the lowest scores will be dropped.
 
Program (enumerated by weeks, all references to Do Carmo’s book):
  1. 1) Chapter 1, Sections 1-2, 1-3: Regular curves, arclength, examples.
  2. 2) Chapter 1, Sections 1-4, 1-5: Vector product, local theory of curves. Curvature, torsion, Frenet equations.
  3. 3) Chapter 1, Section 1-7. Isoperimetric inequality, Cauchy-Crofton Formula.
  4. 4) Chapter 2, Section 2-2: Regular Surfaces, Inverse images of regular surfaces, examples.
  5. 5) Chapter 2, Section 2-3: change of parameters on surfaces.
  6. 6) Chapter 2, Sections 2-4, 2-5: The tangent plane, the differential of a map, First fundamental form.
  7. 7) Chapter 3, Sections 3-2, 3.3 : Definition, computation and properties of the Gauss’ map. Appendix to Section 3: Self-adjoint Linear Maps and Quadratic forms.
  8. 8) Chapter 4, Section 4-2: Isometries, conformal maps.  
  9. 9) Chapter 4, Section 4-3: Theorema Egregium by Gauss.
  10. 10) Chapter 4, Sec. 4-4: Parallel transport and geodesics.
  11. 11) Chapter 4, Sec. 4-5: Theorem of turning tangents, Gauss-Bonnet theorem.
  12. 12) Chapter 4, Sec. 4-6: The exponential map and geodesic polar coordinates.
 
In principle, this finishes requirements for the course according to the course description in the catalogue. We shall use the buffer of additional 2 weeks to either slow down at the difficult parts of the course if necessary, or to study some aspects of differential forms from Flanders’ book.
 
Homeworks (problems with stars are more difficult and are not mandatory):
  1. 1) Sec. 1-2, #1,3; Sec. 1-3 #1, *2, 4,  Due Jan. 26
  2. 2) Sec. 1-4 #1, 5, 13; Sec. 1-5, #1, 9, 11, 12, Due Feb. 2
  3. 3)  Sec.1-7, #3, 6, 10. Due Feb. 9
  4. 4) Sec. 2-2, #1, 3, 7, 8, 12, 16.  Due Feb. 16
  5. 5) Sec. 2-3, #2, 3, 9, 10. Due Feb. 23
  6. 6) Sec. 2-4, #2, 3, 10, 11, *12; Sec. 2-5, #1, 3. Due March 2
  7. 7) Sec. 3-2, #1, 2, 8  Due March 9
  8. 8) Spring break: week March 16-March 21
  9. 9) Sec. 3-3, #1, 5, 7, 12, 18 Due March 23
  10. 10) Sec. 4-2, #1, 4, 5, 12, 16 Due March 30
  11. 11) Sec. 4-3, #1, 2, 3, 5, 8  Due April 6
  12. 12) Sec. 4-4, #5, 14, 15 Due April 13
  13. 13) Sec. 4-5, #2, 3, 4, 5 Due April 20
  14. 14) Sec. 4-6, #1, 2, 3, 4, 7 Due April 27
 
Class cancellations: