Text Elementary Differential Geometry, Andrew Pressley (Springer).
Cornell students have
free electronic access
to this textbook.
This course offers a rigorous and
systematic study of the geometry of lines and
curves in three dimensional Euclidean space and lays the framework for
a more advanced study of differential geometry.
The topics will include: curvature of space curves, regular surfaces
and their parameterizations, the first and second fundamental forms,
the Gauss map,
Gaussian and mean curvature of surfaces, orientability of surfaces,
conformal maps and isometries.
Specifically, we will cover:
chapters 1-4, selection from chapter 5, 6.1-6.3, 7.1-7.4, 8.1-8.2, 9.1-9.2, 9.4, 10.1-10.2, 13.1-13.4.
The material in chapters 1 and 2 will be covered quickly as most is a review of
material from multivariable calculus.
Students will be evaluated both on their ability to do computations related
to the material and to construct proofs relating to the concepts being
studied.
Revised syllabus and schedule
Click here for
a more detailed syllabus.
Click here
for the tentative day-by-day schedule for the semester.
Daily Schedule
Week 13 Final Exam (due 5/16 11:59pm EDT).
- Tuesday (5/5):
the Gauss-Bonnet Theorem for simple closed curves (13.1)
- Thursday (5/7):
the Gauss-Bonnet Theorem for Polygons (13.2); integration on compact surfaces (13.3)
Week 12 Homework assignment 9 (due 5/9 11:59pm EDT).
- Tuesday (4/28):
the Gauss and Codazzi-Mainardi equations (10.1);
Gauss' Theorem Egregium (10.2)
- Thursday (4/30):
surfaces of constant curvature (10.3).
Week 11 Prelim 2 is due on 4/29 at 11:59pm EDT.
- Tuesday (4/21):
geodesics and their equations (9.1, 9.2)
- Thursday (4/23):
geodesics minimize distance (9.4)
Week 10 Homework assignment 8 (due 4/22 11:59pm EDT ).
- Tuesday (4/14):
continue parallel transport and the covariant derivative (7.4); Gaussian and mean curvature (8.1)
- Thursday (4/16):
principle curvatures of a surface (8.2)
Week 9 Homework assignment 7 (due 4/14 11:59pm EDT ).
- Tuesday (4/7):
the second fundamental form (7.1); the Gauss and Weingarten maps (7.2)
- Thursday (4/9):
normal and geodesic curvature (7.3); start parallel transport and the covariant derivative (7.4)
Week 8 Homework assignment 6 (due 4/8 11:00pm EDT).
Starting this week you must upload your assignment via Gradescope.
Click here for some supplemental material on Riemannian metrics and manifolds.
- Tuesday (3/10):
continue with isometries of surfaces (6.2); conformal maps (6.3)
- Thursday (3/12):
conformal maps (6.3)
Week 7 Homework assignment 5 (due 3/10).
- Tuesday (3/3):
surfaces, manifolds and the inverse function theorem (5.6, supplemented)
- Thursday (3/5):
the first fundamental form (6.1); start isometries and surfaces (6.2)
Week 6 Homework assignment 4 (due at noon on 3/4).
- Tuesday (2/25):
Winter break - no class.
- Thursday (2/27):
Examples of surfaces and manifolds (selected material from chapter 5.1-5.4, supplemented)
Week 5
- Tuesday (2/18):
normals and orientability (4.5).
- Thursday (2/20): Prelim 1.
Week 4 Prelim 1 is on 2/20.
Information on prelim 1 can be found here.
Click here for some supplemental material on manifolds in \(\mathbb{R}^n\).
Homework assignment 3 (due 2/18).
- Tuesday (2/11):
continue with surfaces and manfolds (4.1, 4.2, supplemented)
- Thursday (2/13):
smooth maps (4.3), tangents and derivatives (4.4)
Week 3 Homework assignment 2 (due 2/11).
- Tuesday (2/4):
closed curves and Hopf's Umlaufsatz (3.1); the isoperimetric inequality and
Wirtinger's inequality (3.2)
- Thursday (2/6):
the four vertex theorem (3.3); start surfaces and manifolds (4.1, supplemented)
Week 2 Homework assignment 1 (due 2/4).
- Tuesday (1/28):
curvature (2.1) and plane curves (2.2)
- Thursday (1/30):
space curves (2.3)
Week 1
- Tuesday (1/21):
course overview; start chapter 1.
- Thursday (1/23):
curves, arc length, reparameterization, level curves (continue 1.1-1.5)