### Dr. Sarah's Differential Geometry Class Highlights Spring 2009 Page

The following is NOT HOMEWORK unless you miss part or all of the class.
See the main class web page
for ALL homework and due dates.

**Tues Apr 21**
Finish the solutions of the geodesic equations on the sphere by hand.
Geodesics on the cone and the torus in Maple via demos from
John Oprea and Robert
Jantzen.
The geometry of Minkowski space (Lorentzian/Special Relativity). Show
that free particles follow straight line geodesics. Discuss the geometry
of general relativity and Einstein's field equations.

**Thur Apr 23**
Define the potential function and prove the Laplace equation and discuss
Einstein's derivation of the Field equations from his 1916 paper.
Wormhole metric. Discuss the final project.

**Thur Apr 16** Metric form presentations. Solving the
geodesic equations on the sphere by hand.

**Tues Apr 7** Test 2 on Surfaces

**Thur Apr 9** Review test 2. Continue the derivation
of the geodesic equations on a surface with the Christoffel symbols.

**Tues Mar 31** Continue geodesic equations and Christoffel symbols.
Derive the Christoffel symbols on the sphere.

**Thur Apr 2** Take questions on surfaces for the test.
Continue geodesic equations and Christoffel symbols.
Show that geodesics have
constant speed,and derive the geodesic equations with the Christoffel symbols
in the case of surfaces.

**Tues Mar 24** Gauss Curvature and Mean Curvature of a flat
torus and Klein bottle, and in hyperbolic geometry. Geodesic equations
and Christoffel symbols.

**Thur Mar 26** Presentations. Continue geodesic equations and
Christoffel symbols.

**Tues Mar 17**
Applications of the first fundamental form
to surface area calculations using the determinant of the metric form for
the strake, the hyperbolic annulus, and the cone.
Begin curvature.
Normal Curvature 1,
Normal Curvature 2,
Quotations,
Gauss Curvature and Mean Curvature,
helicoid.

**Thur March 19**
Review principal curvatures, Gauss curvature and mean curvatures.
Normal Curvature 1 on a saddle,
Discuss curvatures on a cylinder.
Gauss map of a catenoid. Discuss Gauss map of a cylinder.
Gauss Curvature and Mean Curvature.
Assign project 5. Alone or in groups of 2, prepare a short presentation
for a surface -- Problems: p. 123-126 3.2.11 (hyperboloid of 2 sheets),
3.2.12 (hyperboloid of 1 sheet), 3.2.13 (elliptic paraboloid),
3.2.14 (hyperbolic paraboloid), 3.2.16 (saddle), 3.2.17 (Kuen's Surface),
3.2.19 (Cone), p. 130-132 3.3.3 (torus), 3.3.11 (pseudosphere),
p. 86 2.2.4 (Mobius strip).
Gauss map of a cylinder.
Theorema Egregium of Gauss, that the (Gauss) curvature K of a surface is invariant under isometry. Gauss-Bonnet theorem.

**Tues Mar 3**
Show that distance is exponential in the hyperbolic annulus model.
Begin the first fundamental form and the metric. The plane and
Pythagorean theorem. The helicoid.

**Thur Mar 5**
Review the first fundamental form.
Applications of the first fundamental form
via local isometries (catenoid and helicoid via E,F,G)
deformation.
Examine a saddle and Enneper's surface and use E, F, G to distinguish
them even though they look the same when plotted from u=-1/2..1/2, v=-1/2..1/2.
Additional applications
to surface area calculations using the determinant of the metric form for
the sphere, and the strake.

**Tues Feb 24** Review regular curves. Discuss regular surface
by a hyperboloid. Geodesics on a hyperboloid via symmetry arguments.
Maple file on geodesic and normal curvatures
adapted from David Henderson.
Class commands for the file for
cylinder, sphere, and cone.

**Thur Feb 26**
Maple file on geodesic and normal curvatures
Class commands for the file for
the hyperboloid, cone and helicoid.. Introduction to hyperbolic geometry.

**Tues Feb 17** Go over the first 2 pages of the test.
Define manifolds, orbifolds, surfaces, and geodesics.
Continue with the cylinder.
Use covering arguments to answer questions
about the geodesics. Review extrinsic cylindrical coordinates and define
geodesic rectangular coordinates.

**Thur Feb 19**
Use local coordinates to express the equations of
the geodesics on the cylinder. Clarify the relationship
between curvature and geodesics. Normal to a surface and a regular
patch/parametrization. Normal and geodesic curvature by hand.

**Tues Feb 10**
Finish the isoparametric inequality. Discuss other results from the
global differential geometry of curves.
Review for test 1. If time remains, then begin surfaces.

**Thur Feb 12**
Test 1 on curves.

**Tues Feb 3** Review the Frenet equations. Implications of
the equations. Continue the geometry of helices and torsion/curvature
constant condition. Prove that curvature 0
iff a line. Prove that torsion 0 iff planar.

**Thur Feb 5** Discuss that curvature constant for a plane curve means
part of a circle. Discuss the fundamental theorem of curves.
Discuss results from global differential geometry, including the
isoparametric inequality.

**Tues Jan 27**
Begin applications of the Frenet equations, including
curvature and the strake problem.
Discuss the calculation of curvature when parametrizing by arc length is
impractical.
Discuss and prove the formula for curvature for twice-differentiable
function of one variable.

**Thur Jan 29** Discuss the history of Frenet Frames.
Review Frenet Frames.
Continue deriving the Frenet equations. Begin the geometry of helices.
Maple commands:

with(VectorCalculus):

helix:=<r*cos(t), r*sin(t), h*t>

TNBFrame(helix,t)

simplify(Curvature(helix,t))

simplify(Torsion(helix,t),trig)

**Tues Jan 20** Mention solutions to Calc 3 review on ASULearn.
Prove that the shortest distance between two points is a line. Take
questions on 1.1. Begin 1.2 and 1.3 on arc length and Frenet frames.

**Thur Jan 22**
Begin deriving the Frenet equations.

**Tues Jan 13** Fill out
information sheet.
Introductions. Introduction to the course.
Why is a line the shortest distance path between 2 points?
Our intuition might be that a curve is inefficient since it starts off
pointing away from the endpoint. However this intuition is false on a sphere.
Prove that a line in R^{3} is the shorter than such a curve.
If time remains, begin working on Calc 3 review problems.

**Thur Jan 15** Students present problems from calc 3 review.