Description:

Some of the topics we may cover include:

- Differential manifolds, examples

- Immersion and embedding, examples, introduction to general problem

- Vectorfields, brackets, integrability

- Riemannian metric, connections, differentiation

- Geodesics, convex neighborhoods, exponential map

- Curvature: Riemannian, sectional, Ricci, scalar
- Jacobi fields along geodesics

- Isometric immersion, second fundamental form

- Completeness, Hopf-Rinow and Hadamard theorems

- Constant (sectional) curvature spaces
- Positive curvature results: Bonnet-Myers diameter estimate

- Rauch's comparison estimates on curvature and length

- Morse index theorem for geodesics

- Existence of closed geodesics
- Differential forms and DeRham cohomology

and curves and surfaces.

**Meets:** TTh 9:25AM-10:40AM in
Herman
Brown 423

**Instructor:** Robert Hardt, Herman Brown 430;
Office hours: 11-12 MWF (and others by appt.),

Email: hardt@rice.edu, Telephone: ext 3280

**Homepage:**
http://math.rice.edu/~hardt/401F03/

**Text: ** M.P. doCarmo, *Riemannian
Geometry,* Birkhauser, 1992.

(Also useful is M.P. doCarmo, *Differential Geometry of Curves
and Surfaces,* Prentice-Hall, 1976.)

Handouts:

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 2^{1/2 }(pi). Hint:

find a specific curve in SO(3) between these rotations
whose length is 2^{1/2 }(pi) .

*Homework 2, due. Thurs.,Jan.29* -- Do Carmo page 31-32, #
1(a)(b), #2.

Extra Exercise. Suppose that M is the set of zeros
of a smooth real-valued function f on R^{3}
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/v^{2} times the standard R^{2}
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)/(x^{2}+y^{2}) 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

Solutions
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.