# MATH 402: Differential Geometry, Spring 2004

### Description:

Differential geometry is the study of geometric figures using the methods of calculus.  The principal object is a differentiable manifold, which is roughly a space similar enough to ordinary Euclidean space  Rn  to carry various structures and operations from differential calculus.  This may occur as a submanifold of some  RN , which is simply a subset described locally, after a rotation of  RN , as the graph  { (x,y) :  y = f(x) } corresponding to a smooth RN-n valued function  f  on  Rn .  Whitney showed how an arbitrary  n  dimensional manifold may be identified smoothly with a submanifold of  R2n .  However, there are other useful structures for differential manifolds that are not always compatible with this viewpoint.  Especially important is the notion of a Riemannian metric which provides concepts of length, angle, volume, and many notions of curvature.  There have been many applications.  For example, in general relativity one considers  4 dimensional space-times that may not be globally  R4 or, when accounting for distances, even locally like a region in  R4 . Here gravitational souces contribute  to the curvature of space-time whereas  R4  has no curvature.   Of particular historical significance are the examples of the Riemann unit 2 sphere in  R3  or the Lobaschevski ( hyperbolic) plane which provided geometries showing the failure of Euclid's 5th postulate on the existence or uniqueness of nearby nonintersecting parallel lines.  In a general Riemannian manifold, the notion of a line is replaced by a geodesic which is a curve locally minimizing length or energy.  One can get  an amazingly large number of results by calculations with geodesics, and this will occupy much of our course. Other topics in differential geometry will be chosen based on the interests of the class.

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
Prerequisites for the course include some familiarity with calculus, linear algebra, ordinary differential equations,
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.)

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, # 1(a)(b), #2.
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

### Other beginning books:

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.

• Beltrami
• Blaschke
• Bonnet
• Darboux
• Frenet
• Gauss
• Klein
• Lobachevsky
• Monge
• Riemann
• ### Other Links*:

• Catenoid/Helicoid Deformation
• MapProjections
• Introducing Curves by C. T. J. Dodson
• Introducing Surfaces by C. T. J. Dodson
• Gallery of surfaces
• Dictionaryof special plane curves
• MoreFamous Curves

• *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.