Differential Geometry
The course covers curves and surfaces. After discussing the general setup for curves in n-dimensional Euclidean space, we focus on curve theory in 2 and 3 dimensions. Global problems for closed curves, such as the four vertex theorem, winding number and homotopy classes, and elastic curves will be treated in more detail. In a similar vein, after introducing the general theory of surfaces in 3-space, we discuss special surface classes, such as mininal surfaces (soap films), constant mean curvature surfaces (soap bubbles), and constant Gauss curvature surfaces, in more detail.
Grading

Home work problems will be assigned on a regular basis and graded.
The midterm exam is a take home exam. It will be posted in the week
before spring break and is due in class on Friday March 24, 2017.
The final exam could be project based or a  take home exam.
The total grade will be the equally weighted average of those three grades.
Grader

Kien Nguyen, LGRT 1435 I
kien@math.umass.edu
Available Friday mornings
Text

Christian Baer, Elementary Differential Geometry
Any book with similar title like Elementary Differential Geometry by Barrett O'Neill etc

• hw 1 • hw 2 • hw 3 • hw 4 • hw 5 • hw 6 • midterm • hw 7 • hw 8 • final

 


Course log

Week 1: General facts about curves in n-space
Week 2: Curvature of planar curves
Week 3: Reconstruction of planar curves from curvature
Week 4: Closed curves and winding number
Week 5: Whitney-Graustein Theorem
Week 6: Elastic curves and Calculus of Variations
Week 7: Noether Theorem, characterization of length constrained elastica
Week 8: Parameterized surfaces in 3-space
Week 9: Induced Riemannian metric and abstract (local) RIemannian manifolds
Week10: Differential forms, closed and exact 1-forms, deRham cohomology
Week11: Miracle of Riemannian geometrie--the Levi Civita connection
Week12: Connnection 1-form, Riemann (Gauss) curvature
Week13: Theorema Egregium, fundamental theorem of surface theory (reconstruction from 1st and 2nd fundamental form)