Dr. Sarah's Differential Geometry Class Highlights Spring 2012 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.

  • Thur May 3 Define the potential function and prove the Laplace equation (dot product of the gradient) and discuss the geometry of general relativity and Einstein's field equations from his 1916 paper. Review: torsion and curvature define a curve. Frenet frame can give a basis. For a surface, the first fundamental form and Christoffel symbols tells us local information and we can calculate Gauss curvature from it. We can't just define any E, F and G - need to piece together to give us a nice smooth surface, or one with manageable singularities like the cone. In higher dimensions, the metric has more terms, as do the Christoffel symbols, but they still give us information like the curvature tensor. Einstein's field equations provide a way to solve for "nice" solutions.
  • Tues May 1
    Christoffel symbols and curvature tensor computations for the wormhole metric
    Discuss the geometry of general relativity and Einstein's field equations from his 1916 paper.
  • Thur Apr 26 Finish the
    Geodesics on the cone and the torus in Maple via demos from John Oprea and Robert Jantzen.
    Discuss the homework readings.
    Define spacetime and the Minkowski metric for special relativity. Show that free particles follow straight line geodesics.

  • Tues Apr 24
    Christoffel symbols on the plane and in spherical geometry.
    Discuss the homework readings.
    Geodesics on the cone and the torus in Maple via demos from John Oprea and Robert Jantzen.
  • Thur Apr 19
    Finish deriving the geodesic equations. Use dot product arguments to solve for the Christoffel symbols.
  • Tues Apr 17 Test 2 on Surfaces
  • Tues Apr 12 Take questions on Test 2. Review the Christoffel symbols and begin to derive the geodesic equations.
  • Thur Apr 5 Finish surface presentations. Look at google surface equation search (google plot) Finish curvature of the flat Klein bottle. Begin equations of geodesics. Prove that a geodesic has constant speed. Define the Christoffel symbols and the geodesic equation.
  • Tues Apr 3 Hyperbolic geometry annular model. Gauss curvature K of the annular model, the flat torus and the flat Klein bottle. Gauss-Bonet and angle sum theorems as applications of curvature. Clicker question for your surface:
    a) 0 Gauss curvature K
    b) K < 0
    c) K > 0
    d) More than one answer applies on my surface
    e) Did not complete
    Surface presentations
  • Thur Mar 29
    Clicker questions
    Review Gauss curvature.
    Continue hyperbolic annulus model and showing that distance is exponential. Use this to explore E, F, G and surface area of two geodesics bounded by a horocycle [r times the length of the horocycle base].
    crochet model
  • Tues Mar 27
    Review SA on a sphere: Applications of the first fundamental form:
    Normal Curvature 1, Normal Curvature 2.
    Maximum and mimimum normal curvatures k1 and k2 at a point (principal curvatures). GC=product, mean curvature is the average. Gauss' Theorem egregium: GC is intrinsic quantity.
    Gauss Curvature and Mean Curvature on the sphere.
    GC isometric constant curvature 1 surfaces
    quotations
    clicker questions
    Begin hyperbolic geometry. Show that distance is exponential in the hyperbolic annulus model. Surface area in hyperbolic geometry. Gauss curvature in hyperbolic geometry.
    A special case of the Gauss-Bonet Theorem: Sum of the angles in a triangle equals pi + surface integral over the triangle of the Gauss curvature dA.
  • Thur Mar 22 Research day.
  • Tues Mar 20
    Review surface normal and the first fundamental form. Mention the second fundamental form and Gauss and mean curvature of a surface.
    Look at a deformation of the catenoid and helicoid and EFG of them.
    http://virtualmathmuseum.org/Surface/helicoid-catenoid/helicoid-catenoid.mov
    Applications of the first fundamental form:
    EFG for catenoid and helicoid.
    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.
    Assign homework 6 surfaces and related book pages
    p. 74
    2.1.14 (helicoid)
    2.1.16 (Enneper's Surface)
    p. 80 2.2.4 (Mobius strip)
    p. 114-117:
    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)
    p. 120-121:
    3.3.9 (pseudosphere)
    p. 218 color picture (ellipsoid)
    p. 170 (Scherk's Fifth Surface)
    3.2.19 (Cone)
    3.3.2 (torus)
    Surface area and relationship to the determinant of the metric form
    Surface area on a cylinder and strake and an intrinsic circular disk of radius r on a sphere of radius R.
  • Thur Mar 8
    Proof for the next test: prove that a geodesic must be a great circle.
    Continue with E, F, G and the first fundamental form, and the metric form (ds/dt)2 for the plane (compare with the Pythagorean theorem) and the strake.
    Review the cone spiral geodesic and the 180 degree cone and hand out a covering model. Take questions on the solutions.
  • Tues Mar 6 Continue with the
    Maple file on geodesic and normal curvatures adapted from David Henderson.
    Cone Cone parameterization
    g := (x,y) -> [x*cos(y), x*sin(y), x]:
    a1:=0: a2:=3: b1:=0: b2:=3:
    c1 := 0: c2 := 1:
    Point := 1/2:
    f1:= (t) -> 1/2:
    f2:= (t) -> t:
    latitude circle - discuss why it is not a geodesic using intrinsic arguments, including the lack of half-turn symmetry and the fact that it unfolds to circle.

    Next change to:
    f1:= (t) -> t:
    f2:= (t) -> 1/2:

    Then to:
    b2:=Pi/2:
    cc:=.8497104921: dd:=-.5553603670:
    f1:= (t) -> cc*sec(t/sqrt(2)+dd):
    f2:= (t) -> t:
    Discuss where secant comes from and where cc and dd come from.

    Clicker questions on cones.

    Sphere
    Symmetry arguments on a sphere.
    Parametrization of a sphere. Explain the role of the parameters.
    Normal to the surface, curvature of a latitude, normal and geodesic curvature, E, F, G and the first fundamental form.
    Sphere latitude:
    g := (x,y) -> [cos(x)*sin(y), sin(x)*sin(y), cos(y)]:
    a1:=0: a2:=Pi: b1:=0: b2:=Pi:
    c1 := 1: c2 := 3:
    Point := 2:
    f1:= (t) -> t:
    f2:= (t) -> 1:

    Sphere longitude:
    g := (x,y) -> [cos(x)*sin(y), sin(x)*sin(y), cos(y)]:
    a1:=0: a2:=Pi: b1:=0: b2:=Pi:
    c1 := 1: c2 := 3:
    Point := 2:
    f1:= (t) -> 1:
    f2:= (t) -> t:
  • Thur Mar 1 Review the geodesics on a cylinder. We wrote equations for them intrinsically and extrinsically by examining various coordinate systems on the cylinder:
    rectangular coordinates - (horizontal distance along a base circle, vertical z)
    geodesic polar coordinates - (angle between base circle and geodesic on the cylinder, the arc length of the geodesic on the cylinder)
    extrinsic coordinates - (rcos(theta), rsin(theta), z), with r the radius of a base circle in R3, theta the angle made while traveling on a circle in R3, and z the height on the axis of the cylinder.
    1st and 3rd coordinates: z same, r*theta=horizontal distance along base circle
    Review k=0 iff the curve is a line
    Review the geodesics on a cylinder from symmetry arguments/covering arguments (unwrapping argument from the cover of the plane upstairs to the quotient downstairs where points are glued together to form the cylinder).
    Look at extrinsic equations of the shortest geodesic between two points. Calculate the normal to a surface. Calculate the curvature vector, the projection onto the normal (the normal curvature) and the difference between these vectors (the geodesic curvature - how different from being straight). (calculation of geodesic curvature extrinsically using (cos(u), sin(u), v), normal to surface, the unit tangent vector, and dT/ds.)
    Next examine David Henderson's Maple file:
    Maple file on geodesic and normal curvatures adapted from David Henderson.
    g := (x,y) -> [cos(x), sin(x), y]:
    a1:=0: a2:=2*Pi: b1:=0: b2:=Pi:
    c1 := 1: c2 := 3:
    Point := 2:
    f1:= (t) -> t:
    f2:= (t) -> sin(t):

    The yellow curve does not feel straight since the geodesic curvature (the orange vector) is felt as a turning movement.
    Examine geodesics on a cone and on the sphere in this Maple file.
  • Tues Feb 28 Finish
    Clicker questions on the reading. Extrinsic cylindrical coordinates via extending the inner helix of the strake along the cylinder instead of outwards, the equation of the cylinder, and define geodesic rectangular coordinates. x^2+y^2=1, coordinates u, v, geodesic polar coordinates, (cos(u), sin(u), v). Writing the equations of geodesics.
    Google "History of a cylinder"
    Earliest Known Uses of Some of the Words of Mathematics (C)
  • Thur Feb 23 Build a model as follows:
    1. Turn the paper sideways so that the long side of the paper is horizontal.
    2. Label a point A on the middle of the left (short side) boundary of your (sideways) paper
    3. Fold the paper in half vertically (short folds), parallel to the boundary that has A on it.
    4. Fold the paper in half again vertically (short folds), parallel to the boundary that has A on it.
    5. Unfold and draw dotted lines along the folds. You will have divided the paper into 4 sections and have drawn 3 dotted lines.
    6. Label a point B a bit above and to the right of A, but still in the same section that A is in.
    continue answering cylinder questions
    history including as a surface of revolution and in applications
    Surface area of a cylinder 2 pi r h and derivation of the formula
    Volume of a cylinder p r^2 h and derivation of the formula
    Clicker questions on the reading.
  • Tues Feb 21 Prove the isoparametric inequality. Mention other results from the global differential geometry of curves. Discuss a parametrization of the strake to motivate surfaces. Define manifolds, orbifolds, surfaces, and geodesics. Continue with the cylinder. Use covering arguments to answer questions about the geodesics.
  • Thur Feb 15 Test 1 on curves
  • Tues Feb 13 Clicker questions after hw 3. Put up #2 by hand. Take questions on test 1.
  • Thur Feb 8 Discuss that non-zero curvature constant for a plane curve means part of a circle. Discuss contant torsion curves. Discuss the fundamental theorem of curves for the plane and R^3. Discuss results from global differential geometry, including the isoparametric inequality. Review Green's theorem from calculus 3.
  • Tues Feb 6 Should the Frenet Frame be named after Frenet? Justify your response.
    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 2
    Finish going over #1 on the homework. Warehouse 13
    with(plots);
    plot([(t+t^3)/(1+t^4), (t-t^3)/(1+t^4), t = -10 .. 10]);
    ArcLength(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4)>, t = -10 .. 10)
    Simplify(Curvature(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4)>))
    Torsion(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4),0>)
    TNBFrame(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4),0>)
    Discuss and prove the formula for curvature for a twice-differentiable function of one variable in the form y=f(x).
    Introduction to the Course
    Begin the geometry of helices and applications. 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)
    spacecurve({[5*cos(t), 5*sin(t), 3*t, t = 0 .. 7]})
    Curvature and the strake problem.
    Curvature of a plane curve y=f(x).
  • Tues Jan 31 Collect homework. clicker questions. Mention solutions on ASULearn.
    Fill in the blank for formulas
    Continue deriving the Frenet equations.
    How to calculate T and K when we can't solve for s.
    Discuss curves from #1 on the homework
  • Thur Jan 26 Finish discussing the homework readings. Review via clicker questions. Continue 1.3, including B and the torsion.
  • Tues Jan 24 Register the i-clickers. Finish going over the calc 3 review. Begin 1.2 and 1.3 on arc length and Frenet frames, including jerk and higher time derivatives, T(s), why it is a unit vector, the curvature vector and the magnitude as a scalar, and why the curvature vector is perpendicular to T(s). Discuss the curvature of a circle or radius r (1/r) and the osculating circle. Define the normal vector N. Mention the applets on the main web page. Discuss the homework readings.
  • Thur Jan 19 Finish the proof that a line in R3 is shorter than a curve between 2 points. clicker questions Students present problems from calc 3 review. Look at solutions on ASULearn. work.
  • Tues Jan 17 Fill out information sheet. Introductions. Paramatrized curves in space. Prove that alpha is a curve iff the acceleration is 0. 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 R3 is shorter than such a curve. Ideas from multivariable calculus:
    equation of a line
    tangent line
    equation of a plane
    tangent plane
    parametrizations of curves and surfaces
    velocity, tangent and arc length of a curve
    surface area and volume
    cylindrical and spherical coordinates
    derivative of a function of one variable whose range is in R^3
    partial derivatives of a multivariable function
    gradient
    Green's Theorem
    algebra of vectors in R^2 and R^3 (dot product, cross product, adding vectors, multiplying a vector by a scalar) and geometric representations of these


    Ideas from linear algebra:
    matrix notation a_ij
    addition, multiplication and determinant of 2x2 and 3x3 matrices
    symmetric matrix (A=Transpose(A))
    linear combination of vectors
    span of vectors
    basis of a space
    dimension of a space
    norm of a vector