Some of the topics we may cover include:
Meets: TTh 9:25AM-10:40AM in
Instructor: Robert Hardt, Herman Brown 430;
Office hours: 11-12 MWF (and others by appt.),
Email: email@example.com, Telephone: ext 3280
Text: M.P. doCarmo, Riemannian
Geometry, Birkhauser, 1992.
(Also useful is M.P. doCarmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.)
The Inverse and Implicit Function Theorems.
Homework: This will be assigned
and collected roughly weekly.
Homework 1, due. Thurs.,Jan.22 -- Do Carmo page 32, # 4, 5, 6.
Extra Exercise. Let d(a,b,c) denote the 3 x 3 diagonal matrix with entries a,b,c . In SO(3) show that the
intrinsic distance between the rotations corresponding to d(1,1,1) and d(-1,-1,1) is at most 21/2 (pi). Hint:
find a specific curve in SO(3) between these rotations whose length is 21/2 (pi) .
Homework 2, due. Thurs.,Jan.29 -- Do Carmo page 31-32, #
Extra Exercise. Suppose that M is the set of zeros of a smooth real-valued function f on R3 and that the gradient of f does not vanish anywhere on M .
1) Using the inverse or implicit function theorem show that M is a smooth 2 dimensional manifold.
Corrected Problem 2) Show that if M is a bounded set,
then the one partial derivative df/dx
must vanish at some point of M
Homework 3, due. Thurs.,Feb.5 Do Carmo page 33, #
9(a)(b)(c), page 46,#2
Homework 4, due. Tues.,Feb.12 Do Carmo page 46, #
4(a)(b), page 45,#1
Some hints for #4: First check that the composition of affine
functions f(t) = vt + u and g(t) = yt + x is
(vy)t + (vx+u) so that the group operation on the upper half lane G is given by (u,v)*(x,y) = (vx+u,vy) ..
The identity is (0,1). The differential of the rescaling map (x,y) -> (vx,vy) is simply v times the 2 x 2 identity matrix.
Since we want the unique left invariant metric under this group action, we simply define the metric on G at a point
(u,v) to be 1/v2 times the standard R2 metric. This is also invariant under left and right translation,
i.e. (x,y) -> (x+u,y) . So this is the invariant metric. We now have invariance under all maps (x,y) -> (vx,vy) and
(x,y) -> (x+u,y) . We similarly check that (x,y) -> (-x,y)/(x2+y2) preserves this inner product. This corresponds to the complex map z -> -1/z . These 3 maps generate all the maps z -> (az+b)/(cz+d) with ad-bc = 1 .
Feb.12-19. There will be no new homework this week. Take this
opportunity to catch up on the old homework and to attend the
undergraduate mathematics conference this weekend.
Homework 5, due
Feb.26. P. 56,#1, P.57 #5, p.58, #8(a)
Homework 6, due
Mar.25. P. 77,#1(a)(b), P.83 #7, 8, 9
Homework 7, due April 6. P.83,#7 (Hint: Parallel transport an orthonormal frame from p along radial geodesics.), P.106,#7, P.108, #10.
Projects: Some topics to consider for your
presentation. Complete proofs will not be necessary but your
exposition should give precise definitions, statements, and some
examples. Please discuss you choice with me and I will point you
to some references.
Minkowski Space: relations to special relativity
Differential geometric models for general relativity, Einstein equations
Differential forms - DeRham Theorem
Harmonic forms - Hodge Theorem
Some equations from classical integral geometry
Whitney embedding and immersion theorem for smooth manifolds
Nash isometric embedding theorem for Riemannian manifolds
Computational Differential Geometry
to the Final Exam for Math 401, Fall 2003.
A. Gray, Modern Differential Geometry of Curves and Surfaceswith Mathematica, CRC, 1997.
H. Guggenheimer, Differential Geometry, Dover, 1997.
D. Henderson, Differential Geometry: A Geometric Introduction, Prentice-Hall,1997.
J. Lee, Riemannian Manifolds, An
Introduction to Curvature, Springer, 1991.
R. Millman and G. Parker, Elements of Differential Geometry ,Prentice-Hall, 1997.
B. O'Neill, Elementary Differential Geometry, 2nd Ed.,
AcademicPress, New York, 1997.
B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, , AcademicPress, New York, 1983.
J. Oprea, Differential Geometry and its Applications, Prentice-Hall, 1996.
D. Struik, Lectures on Classical Differential Geometry, Dover, 1988.
*Thanks to Kevin Scannell for these links. Students are welcome to suggest others.
This page is maintained by Robert Hardt ( email)
Last edited 3/18/04.